CONTRIBUTED
P A P E R
Modeling of Nanoscale Devices
Devices and components subject to quantum and atomistic effects, such as
layered semiconductor structures, nanoscale transistors, carbon nanotubes and
nanowires may be modeled using quantum analysis and simulation methods.
By M. P. Anantram, Mark S. Lundstrom, Fellow IEEE , and
Dmitri E. Nikonov, Senior Member IEEE
ABSTRACT | We aim to provide engineers with an introduction
I. INTRODUCTION
to the nonequilibrium Green’s function (NEGF) approach, which
Semiconductor devices operate by controlling the flow of
electrons and holes through a device, and our understanding of charge carrier transport has both benefited from and
driven their development. When Shockley wrote, BElectrons
and Holes in Semiconductors[ [22], semiconductor physics
was at the frontier of research in condensed matter physics.
Over the years, the essential concepts were clarified and
simplified into the working knowledge of device engineers.
The treatment of electrons and holes as semiclassical
particles with an effective mass was usually adequate.
Electronic devices were made of materials (e.g., silicon,
gallium arsenide, etc.) with properties (e.g., bandgap,
effective mass, etc.) that could be looked up. For most
devices, the engineer’s drift-diffusion equation provided a
simple but adequate description of carrier transport. Today
things are changing. Device dimensions have shrunk to the
nanoscale. The properties of materials can be engineered by
intentional strain and size effects due to quantum confinement. Devices contain a countable number of dopants and
are sensitive to structure at the atomistic scale. In addition to
familiar devices like the metal–oxide–semiconductor fieldeffect transistor (MOSFET), which have been scaled to
nanometer dimensions, new devices built from carbon
nanotubes, semiconductor nanowires, and organic molecules
are being explored. Device engineers will need to learn to
think about devices differently. To describe carrier transport
in nanoscale devices, engineers must learn how to think
about charge carriers as quantum mechanical entities rather
than as semiclassical particles, and they must learn how to
think at the atomistic scale rather than at a continuum one.
Our purpose in this paper is to provide engineers with an
introduction to the nonequilibrium Green’s function
(NEGF) approach [5], [8], [9], which provides a powerful
conceptual tool and a practical analysis method to treat small
electronic devices quantum mechanically and atomistically.
We first review the basis for the traditional, semiclassical
description of carriers that has served device engineers for
more than 50 years in Section II. We then describe why this
is a powerful conceptual tool and a practical analysis method to
treat nanoscale electronic devices with quantum mechanical
and atomistic effects. We first review the basis for the
traditional, semiclassical description of carriers that has served
device engineers for more than 50 years. We then describe why
this traditional approach loses validity at the nanoscale. Next,
we describe semiclassical ballistic transport and the Landauer–
Buttiker approach to phase-coherent quantum transport.
Realistic devices include interactions that break quantum
mechanical phase and also cause energy relaxation. As a
result, transport in nanodevices is between diffusive and phase
coherent. We introduce the NEGF approach, which can be used
to model devices all the way from ballistic to diffusive limits.
This is followed by a summary of equations that are used to
model a large class of structures such as nanotransistors,
carbon nanotubes, and nanowires. Applications of the NEGF
method in the ballistic and scattering limits to silicon
nanotransistors are discussed.
KEYWORDS
|
Electron transport; Green’s function; nanoelec-
tronics; nonequilibrium; phonons; quantum mechanics; scattering; semiconductors; simulation; transistor
Manuscript received February 23, 2007; revised April 8, 2008. M. S. Lundstrom was
supported by the National Science Foundation through the Network for Computational
Nanotechnology and by the semiconductor industry through the Semiconductor
Research Consortium, the Focus Center on Materials, Structures, and Devices, and the
Nanoelectronics Research Initiative. M. P. Anantram was supported by NASA Ames
Research Center and National Institute of Standards and Technology (Gaithersburg).
M. P. Anantram is with the Nanotechnology Program, Electrical and Computer
Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada
(e-mail: anant@uwaterloo.ca).
M. S. Lundstrom is with the Department of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 49097 USA
(e-mail: lundstro@purdue.edu).
D. E. Nikonov is with the Intel Corporation, Santa Clara, CA 95052 USA
(e-mail: dmitri.e.nikonov@intel.com).
Digital Object Identifier: 10.1109/JPROC.2008.927355
0018-9219/$25.00 2008 IEEE
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
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Anantram et al.: Modeling of Nanoscale Devices
traditional approach loses validity at the nanoscale. Next, we
describe semiclassical ballistic transport in Section III and
the Landauer–Buttiker approach to phase-coherent quantum
transport in Section IV. Realistic devices include interactions
that break quantum mechanical phase and also cause energy
relaxation. As a result, transport in nanodevices is between
diffusive and phase-coherent. We introduce the NEGF
approach, which can be used to model devices all the way
from ballistic to diffusive limits, in Section V. This is
followed by a summary of equations that are used to model a
large class of structures such as nanotransistors, carbon
nanotubes, and nanowires in Section VI. An application of
the NEGF method in the ballistic and scattering limits to
silicon nanotransistors is discussed in Sections VII and VIII,
respectively. We conclude with a summary in Section IX.
The Dyson’s equations and algorithms to solve for the
Green’s functions of layered structures developed in [23] are
presented in Appendixes II and III. These appendixes can be
skipped by the reader whose aim is to gain a basic
understanding of the NEGF approach to device modeling.
where EðkÞ describes the band structure of the semiconductor. The right-hand side of (2) is simply the velocity of
the semiclassical particle, and in the simplest case it is just
hk=m . By solving (1) and (2), we trace the trajectory of a
carrier in phase space as shown in Fig. 1.
Equations (1) and (2) describe the ballistic transport of
semiclassical carriers. In practice, carriers frequently
scatter from various perturbing potentials (defects, ionized
impurities, lattice vibrations, etc.). The result is that
carriers hop from one trajectory in phase space to another
as shown in Fig. 1. The average distance between
scattering events, the mean-free-path l, has (until recently)
been much smaller than the critical dimensions of a
device. Carriers undergo a random walk through a device
with a small bias in one direction imposed by the electric
field. To describe this scattering-dominated (so-called
diffusive) transport, we should add a random force
ðFS ð~
r; tÞÞ to the right-hand side of (1)
dh~k
¼ rr EC ð~
r; tÞ þ FS ð~
r; tÞ:
dt
II. SEMICLASSICAL TRANSPORT: DIFFUSIVE
Electrical engineers have commonly treated electrons as
semiclassical particles that move through a device under
the influence of an electric field and random scattering
potentials. As sketched in Fig. 1, electrons move along a
trajectory in phase space (position and momentum space).
In momentum space, the equation of motion looks like
Newton’s law for a classical particle
d
h~k
¼ rr EC ð~
r; tÞ
dt
(1)
where ~
k is the crystal momentum and EC is the bottom of
the conduction band. In position space, the equation of
motion is
d~
r 1
¼ rk Eð~
kÞ
dt h
(2)
It is relatively easy to solve (1) and (3) numerically.
One solves the equations of motion (1) and (2) to move a
particle through phase space. Random numbers are chosen
to mimic the scattering process and occasionally kick a
carrier to another trajectory. By averaging the results for a
large number of simulated trajectories, these so-called
Monte Carlo techniques provide a rigorous, though
computationally demanding, description of carrier transport in devices as described in [7].
Device engineers are primarily interested in average
quantities such as the average electron density, current
density, etc. (There are some exceptions; noise is important
too.) Instead of simulating a large number of particles, we
can ask: what is the probability that a state at position ~
r,
with momentum h~
k, is occupied at time t? The answer is
given by the distribution function, f ð~
r; ~
k; tÞ, which can be
computed by averaging the results of a large number of
simulated trajectories. Alternatively, we can adopt a
collective viewpoint instead of the individual particle
viewpoint and formulate an equation for f ð~
r; ~
k; tÞ. The
result is known as the Boltzmann transport equation (BTE)
@f
q~
E
^
þ~
v rr f rk f ¼ Cf
@t
h
Fig. 1. Carrier trajectories in phase space showing free flights along a
trajectory interrupted by scattering events that begin another free
flight. p is momentum and x is position.
1512
(3)
(4)
^ describes the effects of
where ~
E is the electric field and Cf
scattering. In equilibrium, f ð~
r; ~
k; tÞ is simply the Fermi
function, but in general, we need to solve (4) to find f .
Once f ð~
r; ~
k; tÞ is known, quantities of interest to the device
engineer are readily found. For example, to find the
average electron density in a volume centered at
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
position ~
r, we simply add up the probability that all of the
states in are occupied and divide by the volume
nð~
r; tÞ ¼
1X
f ð~
r; ~
k; tÞ:
k
(5)
term minus the divergence of the electron flux) or if
carriers are being generated there. Recombination causes
the electron density to decrease with time. Any physical
quantity must obey a conservation law like (9).
The equation for the first moment of f ð~
r; ~
k; tÞ gives the
equation for average current density [(6)] projected on the
x-axis
Similarly, we find
Current Density
1X
~
Jð~
r; tÞ ¼
ðqÞ~
vf ð~
r; ~
k; tÞ
k
Kinetic Energy Density
1X ~
Wð~
r; tÞ ¼
EðkÞf ð~
r; ~
k; tÞ
k
Energy Current Density
1X ~
~
JE ð~
r; tÞ ¼
EðkÞ~
vf ð~
r; ~
k; tÞ
k
@Jnx 2q dWxx nq2
Jnx
¼
þ Ex :
m dx
@t
m
m
(6)
(7)
@nð~
r; tÞ
¼ rr Fn þ Gn Rn
@t
(9)
where Fn is the electron flux
Jn ¼ qFn :
Each term on the right-hand side of (11) is analogous to the
corresponding terms in (9). The current typically changes
slowly on the scale of the momentum relaxation time m
(of order of subpicosecond time) so the time derivative can
be ignored and (11) solved for
2 dWxx
Jnx ¼ nqn Ex þ n
3
dx
(8)
where q is the absolute value of the electron charge. This
approach provides a clear and fairly rigorous description
of semiclassical carrier transport, but solving the sixdimensional BTE is enormously difficult. One might ask if
we cannot just find a way to solve directly for the
quantities of interest in (5)–(8). The answer is yes, but
some simplifying assumptions are necessary.
Device engineers commonly describe carrier transport
by a few low-order moments of the Boltzmann transport
equation (4). A mathematical prescription for generating
moment equations exists, but to formulate them in a
tractable manner, numerous simplifying assumptions are
required [13]. Moment equations provide a phenomenological description of transport that gives insight and
quantitative results when properly calibrated.
The equation for the zeroth moment of f ð~
r; ~
k; tÞ gives
the well-known continuity equation for the electron
density nð~
r; tÞ
(10)
Gn the electron generation rate and Rn the electron
recombination rate. Equation (9) states that the electron
density at a location increases with time if there is a net
flux of electrons into the region (as described by the first
(11)
(12)
where
n ¼
qm
m
(13)
is the electron mobility and we have assumed equipartition
of energy so that Wxx ¼ W=3, where W is the total kinetic
energy density. This assumption can be justified when
there is a lot of isotropic scattering, which randomizes the
carrier velocity. Equation (12) is a drift-diffusion equation;
it says that electrons drift in electric fields and diffuse
down kinetic energy gradients. Near equilibrium
3
W ¼ nkB T
2
(14)
so when T is uniform, (12) becomes
Jnx ¼ nqn Ex þ kB Tn
dn
dn
¼ nqn Ex þ qDn
dx
dx
(15)
the drift-diffusion equation. By inserting (15) in the
electron continuity equation (9), we get an equation for
the electron density that can be solved for the electron
density within a device. This is the traditional and still
most common approach for describing transport in
semiconductor devices [18].
Since most devices contain regions with high electric
fields, the assumption that W ¼ 3nkB T=2 is not usually a
good one. The carrier energy enters directly into the
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Anantram et al.: Modeling of Nanoscale Devices
Fig. 2. The average velocity (v) versus electric field (E) for electrons in
bulk silicon at room temperature.
second term of the transport equation (12) but also enters
indirectly because the mobility is energy dependent. To
treat transport more rigorously, we need an equation for
the electron energy.
The second moment of f ð~
r; ~
k; tÞ gives the carrier energy
density Wð~
r; tÞ according to (8). The second moment of the
BTE gives a continuity equation for the energy density [13]
@W
dJW
W W0
¼
þ Jnx Ex @t
dx
E
(16)
where W0 is the equilibrium energy density and E the
energy relaxation time. Note that the energy relaxation
time is generally longer than the momentum relaxation
time because phonon energies are small, so that it takes
several scattering events to thermalize an energetic carrier
but only one to randomize its momentum.
To solve (16), we need to specify the energy current.
The third moment of f ð~
r; ~
k; tÞ gives the carrier energy flux
JW ð~
r; tÞ according to (8). The third moment of the BTE
gives a continuity equation for the energy flux
JW ¼ WE Ex þ
dðDE WÞ
dx
(17)
where E and DE are appropriate energy transport mobility
and diffusion coefficient [13].
Equations (9) and (15)–(17) can now be solved selfconsistently to simulate carrier transport. Fig. 2 sketches
the result for uniformly doped, bulk silicon with a constant
electric field. At low electric fields, W 3nkB T=2, and we
find that hvx i n Ex . For electric fields above 104 V/cm,
the kinetic energy increases, which increases the rate of
scattering and lowers the mobility so that at high fields, the
velocity saturates at 107 cm/s. In a bulk semiconductor,
there is a one-to-one relation between the magnitude of the
electric field and the kinetic energy, so the mobility and
diffusion coefficient can be parametrized as known
functions of the local electric field. The result is that, for
bulk semiconductors or for large devices in which the
electric field changes slowly, there is no need to solve all
four equations; we need to solve the carrier continuity and
drift-diffusion equations with field-dependent parameters.
Electric fields above 10 4 V/cm are common in
nanoscale devices. This is certainly high enough to cause
velocity saturation in the bulk, but in a short, high field
region, transients occur. Fig. 3 illustrates what happens for
a hypothetical situation in which the electric field abruptly
jumps from a low value to a high value and then back to a
low value again. Electrons injected from the low field
region are accelerated by the high electric field, but energy
relaxation times are longer than momentum relaxation
times, so the energy is slow to respond. The result is that
the mobility is initially high (even though the electric field
is high), so the velocity can be higher than the saturated
value shown in Fig. 2. As the kinetic energy increases,
however, scattering increases, the mobility drops, and the
velocity eventually decreases to 107 cm/s, the saturated
velocity for electrons in bulk silicon. The spatial width of
the transient is roughly 100 nm; modern devices
frequently have dimensions on this order, and strong
velocity overshoot should be expected.
Fig. 3. The average steady-state velocity (solid line) and kinetic energy (dashed line) versus position for electrons injected into a short slab of
silicon with low-high-low electric field profile.
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Anantram et al.: Modeling of Nanoscale Devices
The example shown in Fig. 3 demonstrates that it is
better to think of the mobility and diffusion coefficient as
functions of the local kinetic energy rather than the local
electric field. What this means is that (9) and (15)–(17)
should all be solved self-consistently to simulate carrier
transport in small devices. Device simulation programs
commonly permit two options: 1) the solution of (9) and
(15) self-consistently with the Poisson equation using
mobilities and diffusion coefficients that depend on the
local electric field (the so-called drift-diffusion approach)
or 2) the solution of (9) and (15)–(17) self-consistently
with the Poisson equation using mobilities and diffusion
coefficients that depend on the local kinetic energy (socalled energy transport or hydrodynamic approaches).
Solving the four equations self-consistently is more of a
computational burden, but it is necessary for when the
electric field changes rapidly on the scale of a mean-freepath for scattering. Actually, numerous simplifying
assumptions are also necessary to even write the current
and energy flux equations as (15) and (17) [13]. The most
rigorous (and computationally demanding) simulations
(so-called Monte Carlo simulations) go back to the
individual particle picture and track carriers trajectories
according to (2) and (3). Drift-diffusion and energy
transport approaches for treating carrier transport in
semiconductor devices have two things in common: the
first is the assumption that carriers can be treated as
semiclassical particles and the second is the assumption
that there is a lot of scattering. Both of these assumptions
are losing validity as devices shrink.
II I. SE MI CL ASS ICAL T RANS PORT :
B AL L I S TI C
Consider the Bdevice[ sketched in Fig. 4(a), which consists
of a ballistic region attached to two leads. The left lead
(source) injects a thermal equilibrium flux of carriers into
the device; some carriers reflect from the potential
barriers within the device, and the rest transmit across
and enter the right lead (drain). A similar statement
applies to the drain lead. The source and drain leads are
assumed to be perfect absorbers, which means that carriers
impinging them from the device travel without reflecting
back into the device. To compute the electron density,
current, average velocity, etc., within the device, we have
two choices. The first choice treats the carriers as
semiclassical particles, and the Boltzmann equation is
solved to obtain the distribution function f ð~
r; ~
k; tÞ, as
discussed in the previous section. The second choice treats
the carriers quantum mechanically, as discussed in the
next section. In this section, we will use a semiclassical
description in which the local density-of-states within the
device is just that of a bulk semiconductor but shifted up or
down by the local electrostatic potential. This approximation works well when the electrostatic potential does not
vary too rapidly, so that quantum effects can be ignored. To
Fig. 4. Sketch of a ballistic device with two leads that function as
reservoirs of thermal equilibrium carriers. (a) Device and the two
leads. (b) Energy band diagram under equilibrium conditions (VD ¼ 0).
(c) Energy band diagram under bias (VD 9 0).
find how the k-states within the ballistic device are
occupied, we solve the Boltzmann transport equation (4).
Because the device is ballistic, there is no scattering, and
^ ¼ 0. It can be shown [13] that the solution to the BTE
Cf
^ ¼ 0 is a function of the electron’s total energy
with Cf
E ¼ EC ðxÞ þ EðkÞ
(18)
where EC ðxÞ is the conduction band minimum versus
position and EðkÞ is the band structure for the conduction
band. We know that under equilibrium conditions
sketched in Fig. 4(b), the proper function of total energy
is the Fermi function
f ðEÞ ¼
1
E EF
1 þ exp
kB T
(19)
where the Fermi level EF and temperature T are constant
in equilibrium.
Now consider the situation in Fig. 4(c), where a drain
bias has been applied to the ballistic device. Although two
thermal equilibrium fluxes are injected into the device, it
is now very far from equilibrium. Since scattering is what
drives the system to equilibrium, the ballistic device is as
far from equilibrium as it can be. Nevertheless, for the
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Anantram et al.: Modeling of Nanoscale Devices
ballistic device, the relevant steady-state Boltzmann
equation is the same equation as in equilibrium. The
solution is again a function of the carrier’s total kinetic
energy. At the leads, we know that the solution is a Fermi
function, which specifies the functional dependence on
energy. For the ballistic device, therefore, the probability
that a k-state is occupied is given by an equilibrium Fermi
function. The only difficulty is that we have two Fermi
levels, so we need to decide which one to use.
Return again to Fig. 4(c) and consider how to fill the
states at x ¼ x1 . We know that the probability that a k-state
is occupied is given by a Fermi function, so we only need to
decide which Fermi level to use for each k-state. For the
positive k-states with energy above ETOP , the top of the
energy barrier, the states can only have been occupied by
injection from the source, so the appropriate Fermi level to
use is the source Fermi level. Similarly, negative k-states
with energy above ETOP can only be occupied by injection
from the drain, so the appropriate Fermi level to use is the
drain Fermi level. Finally, for k-states below ETOP , both
positive and negative velocity states are populated
according to the drain Fermi level. The negative velocity
k-states are populated directly by injection from the drain,
and the positive k-states are populated when negative
velocity carriers reflect from the potential barrier.
Ballistic transport can be viewed as a special kind of
equilibrium. Each k-state is in equilibrium with the lead from
which it was populated. Using this reasoning, one can
compute the distribution function and any moment of it
(e.g., carrier density, carrier velocity, etc.) at any location
within the device. Fig. 5 shows that computed distribution
function in a ballistic nanoscale MOSFET under high gate
and drain bias [21]. A strong ballistic peak develops as
carriers injected from the source are accelerated in the high
electric field near the drain. Each k-state is in equilibrium
with one of the two leads, but the overall carrier distribution
is very different from the equilibrium Fermi–Dirac distribution. When scattering dominates, carriers quickly lose their
Bmemory[ of which lead they were injected from, but for
ballistic transport, there are two separate streams of carriers:
one injected from the source and one from the drain.
To evaluate the electron density versus position within
the ballistic device, we should compute a sum like (5), but
we must do two sums: one for the states filled from the left
lead and another for the states filled from the right lead
nðxÞ ¼ 2
X
L ðxÞfL ðEÞ þ 2
kL
X
R ðxÞfR ðEÞ
(20)
kR
where fL and fR are the equilibrium Fermi functions of leads
L and R and L;R ðxÞ is a function that selects out the k-states
at position x that can be filled by lead L or R according to the
procedure summarized in Fig. 4. The factors of B2[ in
(20) correspond to summation over spin states. It is often
convenient to do the integrals in energy space rather than in
k-space, in which case (20) becomes
nðxÞ ¼
Z
dE½LDOSL ðx; EÞfL ðEÞþLDOSR ðx; EÞfR ðEÞ
(21)
where LDOSL;R ðx; EÞ is the local density of states at energy E,
fillable from lead L or R. The density of state contains
summation over spin states. For diffusive transport, we deal
with a single density-of-states and fill it according to a source
quasi-Fermi level, but for ballistic devices, the density of
states separates into parts fillable from each lead.
The current flowing from source to drain (drain to
source) lead is simply the transmission probability TðEÞ
times the Fermi function of the source (drain) lead. The
net current flowing in the device is then
I¼
2e
h
Z
dETðEÞ½fL ðEÞ fR ðEÞ:
(22)
For the semiclassical example of Fig. 4, TðEÞ ¼ 0 for
E G ETOP and TðEÞ ¼ 1 for E 9 ETOP .
I V. PHASE COHERENT QUANTUM
TRANSPORT: THE
L A N DA UER–B UTTI KE R FORMAL I SM
Quantum mechanically, the electron is a wave and the wave
function ð~
rÞ is obtained by solving Schrodinger’s equation
Fig. 5. The computed carrier distribution function within a nanoscale
MOSFET under high gate and drain bias. (From [21].)
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Proceedings of the IEEE | Vol. 96, No. 9, September 2008
h2 2
r þ Vð~
rÞ ð~
rÞ ¼ Eð~
rÞ
2m
(23)
Anantram et al.: Modeling of Nanoscale Devices
ðRÞ
ðlocÞ
Fig. 6. All wave functions in the device can be represented as left incident ððLÞ
D Þ, right incident ðD Þ, or localized states ðD Þ.
where E is the energy. Consider a device connected to two
leads as shown in Fig. 6, where the leads are assumed to
have a constant electrostatic potential. In a manner
identical to the discussion of semiclassical modeling in
the previous section, where we considered corpuscular
electrons incident from the left and right leads, in quantum
mechanical modeling, we need to consider electron waves
incident from the left and right leads. The electron wave
function in device region D can be thought to arise from the
following.
• Waves incident from the left lead (L) of the form
eikx , which have transmitted and reflected compo0
nents teik x and reikx in the right and left leads,
respectively. The wave function in the device
ðLÞ
region D due to this wave is represented by D .
More properly, we must attach a subscript k (or E)
to denote which state (or energy level) in the left
ðLÞ
lead induced the state D in the device, but we
leave it out for compactness.
• Waves incident from the right lead (R) of the form
0
eik x , which have transmitted and reflected
0
components t0 eikx and r0 eik x in the left and right
leads, respectively. The wave function in the
device region D due to this wave is represented
ðRÞ
by D . Again k (or E) is implicit.
• States localized in device region D represented by
ðlocÞ
D . Localized and quasi-localized states are
filled up by scattering due to electron–phonon and
electron–electron interaction. We will assume
here that localized states are absent.
The Landauer–Buttiker approach expresses the expectation value of an operator in terms of the left and right
incident electrons from the leads and their distribution
functions. The expectation value of operator Q^ is
Q¼
i
E
D
E
XhD ðLÞ
^ DðLÞ fL ðEÞ þ ðRÞ
^ ðRÞ fR ðEÞ :
D jQj
D jQjD
k;s
(24)
Here the summation is performed over the momentum k
and the spin s states.
The operator Q^ can be just the number 1, in which case
the summation gives the number of occupied states; or it
can be the momentum operator ihðd=dxÞ. The crucial
point here is that we are able to simply add the
contributions from the left and right leads in the absence
of scattering. One may argue that this cannot be done since
electrons are subject to Pauli’s principle and two electrons
injected into the device cannot occupy the same state at
the same time. However, since the Hamiltonian of the
whole system (drain, source, and reservoirs) is Hermitian,
the states injected from the source and the drain are
orthogonal to each other. Since they are now two distinct
states with no overlap, two electrons can occupy them
without violating Pauli’s principle. Another implication of
the orthogonality of the source- and drain-injected states is
ðLÞ
that, instead of taking a linear combination of D and
ðRÞ
^ we can consider
D to extract the expectation values of Q,
them separately (this is not true if we have a new
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Anantram et al.: Modeling of Nanoscale Devices
interaction in the original phase-coherent Hamiltonian H,
like the electron–phonon interaction, which couples a
source-injected state and a drain-injected state).
Since in this paper we do not consider any spindependent phenomena, the spin summation translates
into a factor of two. Equation (24) has contributions from
two physically different sources. The first term corresponds to contribution from waves incident from the left
ðLÞ
lead ðD Þ at energy E, weighted by the Fermi factor of
the left lead ðfL Þ. The second term corresponds to waves
ðRÞ
incident from the right lead ðD Þ weighted by the Fermi
factor of the right lead ðfR Þ. More generally, if device
region D is connected to a third lead G, then the
expectation value of operator Q^ is
Q¼
E
D
E
XhD ðLÞ
ðRÞ ^ ðRÞ
^ ðLÞ
D jQj
fL ðEÞ þ D jQj
fR ðEÞ
D
D
k;s
i
D
E
ðGÞ ^ ðGÞ
þ D jQj
ðEÞ
f
G
D
(25)
ðGÞ
where D corresponds to the wave function in the Device
due to waves incident from lead G and fG is the Fermi
factor of lead G. Using (24), the contribution to electron
ðnÞ and current (J) densities at x in the device region D are
given by (neglecting gate lead)
X ðLÞ 2
ðRÞ 2
nðxÞ ¼
D ðxÞ fL ðEÞ þ D ðxÞ fR ðEÞ
(26)
k;s
and
"
ðLÞ
X e
h
d ðxÞ
ðLÞ
D ðxÞy D
fL ðEÞ
JðxÞ ¼
2mi
dx
k;s
ðRÞ
þ D ðxÞy
ðRÞ
dD ðxÞ
fR ðEÞ c.c
dx
#
(27)
where Bc.c.[ represents complex conjugate of the terms to
its left. The quantum mechanical density of states at
energy E due to waves incident from the left ðLDOSL Þ and
right ðLDOSR Þ are
LDOSL ðx; EÞ ¼ 2
X ðLÞ 2
D ðxÞ
(28)
X ðRÞ 2
D ðxÞ
(29)
kl
LDOSR ðx; EÞ ¼ 2
kr
where kl and kr are states with energy E incident from the
left and right leads, respectively. Then the electron density
1518
can be written in the same form as (21)
nðxÞ ¼
Z
LDOSL ðx; EÞfL ðEÞdE
Z
þ LDOSR ðx; EÞfR ðEÞdE (30)
except that the expressions for the local density of states are
different. Similarly, (27) can be expressed in a form identical
to (22). The above formalism can be extended to calculate
noise (shot and Johnson-Nyquist) in nanodevices [3]. Device
modeling in the phase-coherent limit involves solving
Schrodinger’s equation to obtain the electron density selfconsistently with Poisson’s equation.
V. QUANTUM T RANSPORT WITH
SCATTERI NG: T HE NEED FOR
GREEN’S FUNCTIONS
The description in the previous section is valid only in the
phase-coherent limit. The terminology Bphase coherent[
refers to a deterministic evolution of both the amplitude
and phase of n ð~
rÞ as given by Schrodinger’s equation. The
quantum mechanical wave function evolves phase coherently only in the presence of rigid scatterers, a common
example of which is the electrostatic potential felt by an
electron in the device. The wave function of an electron
loses phase coherence due to scatterers that have an
internal degree of freedom such as phonons. Phaseincoherent scattering involves irreversible loss of phase
information to phonon degrees of freedom. Naturally,
including loss of phase information is important when
device dimensions become comparable to the scattering
lengths due to phonons and other phase-breaking mechanisms. Accurate modeling of nanodevices should have the
ability to capture:
• interference effects;
• quantum mechanical tunneling;
• discrete energy levels due to confinement in twodimensional (2-D) and three-dimensional device
geometries;
• scattering mechanisms (electron–phonon,
electron–electron).
The first three effects can be modeled by solving
Schrodinger’s equation in a rigid potential as discussed
in Section IV. While in the semiclassical device modeling,
the Boltzmann equation accounts for the energy and
momentum relaxation due to scattering mechanisms, in
quantum mechanical device modeling, the NEGF approach
is necessary to account for energy, momentum, and
quantum mechanical phase relaxation.
The semiclassical approaches to transport can be derived
from quantum mechanics [2], [10]. In its current form of
implementation, the quantum mechanical approaches,
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
however, need a considerable amount of work to model a
broad class of realistic devices. We will now briefly discuss
three topics that require additional work. The first topic is
the inclusion of scattering mechanisms between electrons
and phonons and other electrons (including plasmons).
Here, the current implementations of scattering mechanisms in the Monte Carlo method to device modeling are
significantly more comprehensive [6]. While there is no
fundamental roadblock to including these scattering mechanisms, efficient methods of determining the self-energies
and computing the Green’s functions are required to make
quantum transport simulation practical. The second area
that requires further work is a careful understanding of the
coupling between the device and the leads. In most
implementations of quantum transport, it is assumed that
the role of leads can be added as a simple series resistance.
This assumption usually fails unless the leads correspond to
wide regions consisting of a large number of modes
compared to the number of modes in the device. In a
number of practical situations, an abrupt demarcation
between the device and leads does not exist. In such cases,
considerable care must be exerted to determine the
physically optimum demarcation, which can sometimes
lead to computationally intractable problems. The third area
that requires further work is three-dimensional modeling.
Compared to the sophisticated three-dimensional modeling
possible in the drift-diffusion framework, the computational
modeling of quantum transport is in its infancy. The
algorithms required to handle experimentally relevant
three-dimensional nanodevices currently do not exist.
Most work on quantum transport that has yielded insights
into experiments is based on solving the quantum transport
equations in reduced dimensions using detailed device
physics known to experts. The field is far away from the state
where one can define a three-dimensional real space or
tight-binding grid and model experiments.
The NEGF approach is based on the rigorous many-body
quantum theory [16]. It is designed to describe carrier
transport with scattering. In its derivation [4], it makes the
assumptions similar to those of the quantum Boltzmann
equation [or Keldysh–Kadanoff–Baym approach (KKB)] [8]
and [9]; also see [16]. The main assumptions are: i) a single
particle approach and ii) a mean-field approximation. Note
that NEGF goes beyond KKB in treating interactions with
reservoirs, e.g., electric leads, in a manner similar to Landauer
approach; see [5]. When applying the NEGF approach to
devices, additional assumptions are frequently made, such as
the self-consistent Born approximation in the perturbation
theory and the neglect of off-diagonal elements in the
scattering matrices. In practice, the NEGF approach provides
a good description for many devices. When scattering is
strong and potential variations slow, it can be shown to reduce
to the Boltzmann transport equation [15], which forms the
basis for semiclassical modeling of devices. There are,
however, important problems that the NEGF approach
cannot describe. These have to do with strongly correlated
transport in devices that display so-called single electron
charging effects [28].
In the remainder of this section, we explain the NEGF
approach in the phase-coherent limit by starting from
Schrodinger’s equation [5]. We will start by an explanation
of the tight-binding Hamiltonian and relate this Hamiltonian
to a device with open boundary conditions (Section V-A).
The open boundary conditions lead to an infinite dimensional matrix. We will describe a procedure to fold the effect
of the open boundaries into the finite device region in
Section V-B. This will allow us to deal with small matrices
where the open boundaries are modeled by lead self-energies.
The Green’s functions, self-energies, and their relationship
to current and electron density are derived in Section V-C.
Then in Section V-D, we extend the discussion in Section V-C
to include electron–phonon interaction, which is where
the NEGF approach is really essential.
A. Tight-Binding Hamiltonian for a
One-Dimensional Device
Consider a system described by a set of one-dimensional
(1-D) grid/lattice points with uniform spacing a. Further
assume that only nearest neighbor grid points are coupled.
A spatially uniform system with a constant potential has the
Hamiltonian shown in (31) at the bottom of the page or
tq1 þ ðE Þq tqþ1 ¼ 0
where E is the energy and q is the wave function at grid
point q. The Hamiltonian matrix is tridiagonal because of
nearest neighbor interaction. The diagonal and offdiagonal elements of the Hamiltonian and t represent
the potential and interaction between nearest neighbor
grid points q and q þ 1, respectively.
10
0
B
B
B
t
E
B
ðE HÞ ¼ 0 ! B
B
B
B
@
(32)
t
t E t
t
E t
1
CB C
C
CB
CB q1 C
C
CB
CB q C ¼ 0
C
CB
CB qþ1 C
C
CB
A@ A
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The solution of (32) can be easily verified using Bloch
theorem to be
E ¼ þ 2t cosðkaÞ
(33)
q ¼ eikqa
(34)
and the group velocity is
v¼
1 @E
2at
¼
sinðkaÞ:
h @k
h
•
•
device (D) with an arbitrary potential;
right semi-infinite lead (R) with a constant
potential r .
The potential of the left (right) lead l ðr Þ and the
hopping parameter tl ðtr Þ are assumed to be constant,
which signifies that the leads are highly conducting
and uniform. Then the Hamiltonian of the device and
leads (36) is an infinite dimensional matrix that can
be expanded as
(35)
The uniform tight-binding Hamiltonian in (32) can be
extended to a general nearest neighbor tight-binding
Hamiltonian given by
tq;q1 q1 þ ðE q Þq tq;qþ1 qþ1 ¼ 0
(36)
tl l3 þ ðE l Þl2 tl l1 ¼ 0
tl l2 þ ðE l Þl1 tl;d 1 ¼ 0
(38)
td;l l1 þ ðE 1 Þ1 t1;2 2 ¼ 0
(39)
t2;1 1 þ ðE 2 Þ2 t2;3 3 ¼ 0
(40)
where q is the on-site potential at grid point q and tq;qþ1 is
the Hamiltonian element connecting grid points q and
tqþ1;q ¼ tyq;qþ1 1. tqþ1;q ¼ tyq;qþ1 because the Hamiltonian is
Hermitian.
In the special case of the discretized Schrodinger
equation on a uniform grid
tn;n1 n1 þ ðE n Þn td;r r1 ¼ 0
(41)
tr;d n þ ðE r Þr1 tr r2 ¼ 0
(42)
tr r1 þ ðE r Þr2 tr r3 ¼ 0
2
t ¼ tq;qþ1 ¼ tqþ1;q ¼ 2
h
h
and q ¼ Vq þ 2
2
2ma
ma
(37)
where a is the grid spacing and Vq is the electrostatic
potential at grid point q.
B. Eliminating the Left and Right Semi-Infinite Leads
A typical nanodevice can be conceptually divided into
three regions (Fig. 7):
• left semi-infinite lead (L) with a constant
potential l ;
where the top and bottom bullets represent the semiinfinite left and right leads. The subscript lmðrmÞ refers to
grid point m in the left (right) lead. However, to find the
electron density in (26), the wave function is only required
at the device grid points. We will now discuss a procedure
to fold the influence of the left and right semi-infinite
leads into the device region.
Terminating the Semi-Infinite Left and Right Leads: The
wave function in the leads due to waves incident from the
Fig. 7. A one-dimensional device connected to two semi-infinite leads. While the potential in the leads (L and R) is held fixed, the potential
in the device (D) can vary spatially.
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Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
left lead is
solving the following n-dimensional matrix instead of the
infinite dimensional matrices in (38)–(42)
ln ¼ ðeþikl xln þ sll eikl xln Þ in region L
ikr xrn
rn ¼ srl e
in region R
(43)
ðLÞ
AD ¼ iL
(44)
(50)
ðLÞ
where xln and xrn correspond to integer times grid spacing ðaÞ.
The normalization constant has been neglected in the above
equations. The corresponding eigenvalues are [(33)]
where A is a square matrix of dimension n and D and iL
are n by 1 vectors. iL is the source function at ðk; EÞ due to
the left lead. Matrix A is
E l ¼ 2tl cosðkl aÞ ¼ tl ðeikl a þ eikl a Þ
A ¼ EI HD lead
(45)
and, similarly for the right lead, with the indexes r. sll and srl
are the reflection and transmission amplitudes. Substituting
(43) and (45) in (38) yields
sll ¼ t1
l ðtl þ tld 1 Þ:
and the nonzero elements of lead are
lead1;1 ¼ td;l eþikl a t1
l tl;d ¼ L
leadn;n
¼ td;r eþikr a t1
r tr;d
¼ R :
(47)
Equation (47) is a modification of Schrodinger’s equation
centered at grid point 1 of the device [(39)] to include
the influence of the entire semi-infinite left lead.
Similarly, substituting (44) and E r ¼ 2tr cosðkr aÞ
into (42), we get
eikna
tr;d n :
tr
(53)
L and R are called the self-energies, and they represent
the influence of the semi-infinite left and right leads on the
device, respectively. The real part of self-energy shifts the
on-site potential at grid point 1 from 1 to 1 þ ReðL Þ.
The imaginary part of self-energy multiplied by 2 is the
scattering rate of electrons from grid point 1 of the device
to the left lead (scattering rate ¼ 2Im½L ) in the weak
coupling limit.
In a manner identical to the derivation of (50), for
waves incident from the right lead, the wave function in
ðRÞ
the device ðD Þ can be obtained by solving
ðRÞ
AD ¼ iR
srl ¼
(52)
(46)
Substituting (43) and (46) in (39), we obtain
ðE 1 td;l eþikl a t1
l tl;d Þ1 t1;2 2
¼ 2itd;l sinðkl aÞ:
(51)
(48)
Now, substituting (44) and (48) into (41), we can terminate
the right semi-infinite region to yield
(54)
where iR is the source function due to the right semiinfinite lead. The only nonzero elements of A, iL and iR are
Að1; 1Þ ¼ E1 L and Aðn; nÞ ¼ En R
(55)
Aði; iÞ ¼ Ei ;
eikr a
tn;n1 n1 þ E n td;r
tr;d n ¼ 0:
tr
Aði; i þ 1Þ ¼ ti;iþ1 and Aði þ 1; iÞ ¼ tyi;iþ1
(49)
Equation (49) is a modification of Schrodinger’s equation
centered at grid point n of the device (41) to include the
influence of the entire semi-infinite right lead.
The influence of the semi-infinite left and right leads
have been folded into grid points 1 and n of the device for
waves incident from the left lead [(47) and (49)]. Now the
wave function in the device due to waves incident from the
left lead can be obtained (to within a phase factor) by
iL ð1Þ ¼ 2itd;l sinðkl aÞ
iR ðnÞ ¼ 2itd;r sinðkr aÞ:
(56)
(57)
(58)
C. Electron and Current Densities Expressed in
Terms of Green’s Functions
The Green’s function corresponding to Schrodinger’s
equation ð½E H ¼ 0Þ for the device and leads is
½E H þ iG ¼ I
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where is an infinitesimally small positive number that
pushes the poles of G to the lower half-plane in complex
energy, and H is the Hamiltonian. This Green’s function is
defined in a spirit similar to the Green’s function for
Poisson’s equation. In the rest of this paper, is implicit.
The Green’s function of device region D with the influence
of the leads included is
where
1
jsinðkl aÞjtl;d fL ðEÞ
jtl j
1
in
jsinðkr aÞjtr;d fR ðEÞ:
R ðEÞ ¼ 2td;r
jtr j
in
L ðEÞ ¼ 2td;l
(60)
kl and kr at energy E are determined by E ¼ l þ 2tl cosðkl aÞ
and E ¼ r þ 2tr cosðkr aÞ [(33)], respectively. It can be
seen from (52), (53), (68), and (69)
A ¼ EI HD lead ðEÞ
(61)
in
L ðEÞ ¼ 2Im½L ðEÞfL ðEÞ
(70)
in
R ðEÞ
(71)
is an n-dimensional matrix defined in (55) and (56).
Using the definition for G in (50) and (54), the wave
function in region D due to waves incident from left and
right leads can be written as
ðLÞ
D ¼ GiL
ðRÞ
D ¼ GiR :
(62)
(63)
k;s
¼
X
(64)
Gq;1 4td;l sin2 ðkl aÞtl;d fL ðGy Þ1;q
k;s
þ Gq;n 4td;r sin2 ðkr aÞtr;d fR ðGy Þn;q
(65)
where Gy is the Hermitian conjugate of the Green’s
function. The summation over k can be converted to an
integral over E by,
X
k
!
Z
dE dk :
2 dE
nq ¼ 2
1522
(72)
in
in
lead1;1 ðEÞ ¼ L ðEÞ
(73)
in
leadn;n ðEÞ
(74)
¼ in
R ðEÞ:
Z
dE½LDOSL ðq; EÞfL ðEÞ þ LDOSR ðq; EÞfR ðEÞ (75)
(66)
where LDOSL ðq; EÞ ðLDOSR ðq; EÞÞ is the density of states
due to waves incident from the left (right) lead at grid
point q and
dE h
y
Gq;1 ðEÞin
L ðEÞðG Þ1;q ðEÞ
2
i
y
þ Gq;n ðEÞin
R ðEÞðG Þn;q ðEÞ
dE
y
GðEÞin
lead ðEÞG ðEÞjq;q
2
in
in
L and R defined above in (70) and (71) are called the
in-scattering self-energies due to leads. These self-energies
physically represent in-scattering of electrons from the
semi-infinite leads to the device and so play an important
role in determining the charge occupancy in the device.
They depend on the Fermi–Dirac factor/occupancy in the
leads fL and fR and the strength of coupling between leads
and device Im½L ðEÞ and Im½R ðEÞ.
It is easy to see that the electron density in (67) and
(72) can also be written as
nq ¼
Using (66) and jdE=dkj ¼ 2ajtjj sinðkaÞj (where k 2 kl ,kr
and t 2 tl , tr ), (65) becomes
Z
Z
where the nonzero elements of in
lead are
As iL and iR are nonzero only at grid points 1 and n, the full G
matrix is not necessary to find the wave function in the device;
only the two columns Gð:; 1Þ and Gð:; nÞ are necessary.
The electron density at grid point q can now be written
using (62) and (63) in (26) as
Gq;1 iL iyL ðGy Þ1;q fL þ Gq;n iR iyR ðGy Þn;q fR
¼ 2Im½R ðEÞfR ðEÞ:
The electron density [(67) or (72)] can then be written as
nq ¼ 2
nq ¼
(69)
AG ¼ I
where
X
(68)
LDOSL ðq; EÞ ¼
(67)
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
LDOSR ðq; EÞ ¼
Gq;1 L ðGy Þ1;q
Gq;n R ðGy Þq;n
(76)
(77)
Anantram et al.: Modeling of Nanoscale Devices
where
applied biases can now be written as
L ðEÞ ¼ 2Im½L ðEÞ
R ðEÞ ¼ 2Im½R ðEÞ:
(78)
(79)
Note that (75) is identical to (21) for semiclassical ballistic
transport.
The current density between grid points q and q þ 1 per
unit energy can be written as [(27)]
Jq!qþ1 ðEÞ ¼
e
h h ðLÞ y ðLÞ
ðLÞ y
2 q qþ1 qþ1 ðLÞ
fL ðEÞ
q
2mai
i
y ðRÞ
ðRÞ y ðRÞ
þ ðRÞ
ðEÞ
: (80)
f
R
qþ1
qþ1
q
q
Now, following the derivation for electron density above
[(67)], it is straightforward to derive that the current
density
Z
ie
h
dE
2
Jq!qþ1 ¼
2ma
2
h
y
Gq;1 ðEÞin
L ðEÞðG Þ1;qþ1 ðEÞ
y
þ Gq;n ðEÞin
R ðEÞðG Þn;qþ1 ðEÞ
y
Gqþ1;1 ðEÞin
L ðEÞðG Þ1;q ðEÞ
i
y
ðEÞðG
Þ
ðEÞ
:
Gqþ1;n ðEÞin
R
n;q
(81)
ie
h
2
2ma
Z
dE h
y
GðEÞin
lead ðEÞG ðEÞjq;qþ1
2
(84)
(85)
That is, the diagonal and first off-diagonal elements of Gn
are related to the electron and current densities, respectively. Note that these equations are equivalent to (26) and
(27) appearing in the Landauer–Buttiker approach.
Equations (72) and (84) have to be divided by the grid
spacing a for the density to have units of per unit length.
2) Hole Correlation Function: In the absence of phasebreaking scattering, the Green’s function ðGÞ and the electron
correlation function ðGn Þ are sufficient for device modeling.
Scattering introduces the need for the hole correlation
function ðGp Þ, whose role will become clearer in Section V-D.
While the Gn Green’s function is directly proportional to the
density of occupied states, the hole correlation function is
proportional to the density of unoccupied states.
The density of unoccupied states at grid point q is
also obtained by applying the Landauer–Buttiker formalism. For this, we simply replace the probability of finding
an occupied state in the lead fL;R by the probability of
finding an unoccupied state in the lead 1fL;R in (26).
Then following the derivation leading to (26), we obtain
hq ¼ 2
Z
Z
dE h
y
Gq;1 ðEÞout
L ðEÞG1;q ðEÞ
2
i
y
þ Gq;n ðEÞout
ðEÞG
ðEÞ
R
n;q
dE
y
GðEÞout
lead G ðEÞjq;q
2
(86)
(87)
where the only nonzero elements of out
lead are
i
y
GðEÞin
lead ðEÞG ðEÞjqþ1;q : (82)
1) Electron Correlation Function: More generally, we
define the electron correlation function Gn , which is the
solution to
y
AGn ¼ in
lead G :
Gnq;q ðEÞ
2
i
ieh 1 h n
2
Gq;qþ1 ðEÞ Gnqþ1;q ðEÞ :
Jq!qþ1 ðEÞ ¼
2ma 2
hq ¼ 2
The current density is given by
Jq!qþ1 ¼
nq ðEÞ ¼ 2
(83)
Noting that G ¼ A1 , it is easy to obtain (72) Gn ¼
y
Gin
lead G .
The expressions for electron [(72)] and current [(82)]
densities at energy E in the phase-coherent case at finite
1
jsinðkl aÞjtl;d ð1 fL Þ
jtl j
¼ 2Im½L ðEÞ½1 fL ðEÞ
(88)
1
out
out
jsinðkr aÞjtr;d ð1 fR Þ
leadn;n ðEÞ ¼ R ðEÞ ¼ 2td;r
jtr j
¼ 2Im½R ðEÞ½1 fR ðEÞ:
(89)
out
out
lead1;1 ðEÞ ¼ L ðEÞ ¼ 2td;l
Akin to (83) and (84), the density of unoccupied states
at energy E at grid point q can be expressed as the diagonal
elements of Gp
hq ðEÞ ¼ 2
Gpq;q ðEÞ
2
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Anantram et al.: Modeling of Nanoscale Devices
in
Fig. 8. Pictorial representation of the two in-scattering self-energies that appear in this paper. in
L ðEÞ and R ðEÞ are the self-energies of the leads
ðEÞ
is
the
self-energy
due
to
electron–phonon
interaction and is nonzero at all
and are nonzero only at the first and last device grid points. in
Phonon
device grid points.
where Gp is in general given by
y
AGp ¼ out
lead G :
(91)
Equations (86) and (90) have to be divided by the grid
spacing a for the density to have units of per unit length.
D. Electron-Phonon Scattering
In Section V-C, we defined the self-energies in the
device arising from coupling of the device to the external
leads. The self-energy in
L represents in-scattering of
electrons (in-scattering rate) from the semi-infinite left
lead to the device, assuming that grid point 1 of the
device is empty. A similar statement applies to in
R . The
in-scattering self-energy of the leads depends on their
Fermi distribution functions and surface density of
states.
A second source for in-scattering to grid point q and
energy E is electron–phonon interaction. The self-energy
at ðq; EÞ has two terms corresponding to in-scattering
from ðq; E þ h!phonon Þ and ðq; E h!phonon Þ, as shown in
Fig. 8. Intuitively, the in-scattering self-energy (inscattering rate) at ðx; EÞ should depend on the Bose
factor for phonon occupancy, the deformation potential
for electron–phonon scattering and the availability of
electrons at energies E þ h!phonon and E h!phonon . It
follows rigorously, within the Born approximation, that
the in-scattering self-energy at energy E and grid point q
is [15]
in
Phononq;q ðEÞ ¼
X
h
Dq nB ð
h!phonon ÞGnq;q ðE
h!phonon Þ
i
þ nB ð
h!phonon Þ þ 1 Gnq;q ðE þ h!phonon Þ : (92)
1524
Dq represents the electron–phonon scattering strength at
grid point q due to a phonon mode . The first term of (92)
represents in-scattering to E from E h!phonon (phonon
absorption). nB is the Bose distribution function for
phonons of energy h!phonon and Gnq ðE h!phonon Þ is the
electron density at E h!phonon . The first and second
terms of (92) represent in-scattering of electrons from
E h!phonon (phonon absorption) and E þ h!phonon
(phonon emission) to E, respectively. The in-scattering
rate at grid point q is given by
In-scattering rate at grid point q :
¼ in
q;q ðEÞ
h
qin ðEÞ
(93)
where in
q;q is the sum of all in-scattering self energies at
grid point q.
The out-scattering self-energy out
in (88) represents
L
out-scattering of electrons from grid point 1 in the
device to the semi-infinite left lead, assuming that grid
point 1 of the device was occupied. The out-scattering
self-energy due to the left lead out
depends on the
L
probability of finding an unoccupied state in the left lead
1 fL and the surface density of states of the left lead. A
similar statement applies to out
R . A second source for
out-scattering of electrons from an occupied state at
ðq; EÞ is electron–phonon interaction, which leads to
scattering to ðq; E þ h!phonon Þ and ðq; E h!phonon Þ as
represented in Fig. 9. Intuitively, the out-scattering selfenergy (out-scattering rate) at ðq; EÞ should depend on
the Bose factor for phonon occupancy, the deformation
potential for electron–phonon scattering, and the availability of unoccupied states at energies E þ h!phonon and
E h!phonon . It follows rigorously, within the Born
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
Fig. 9. Pictorial representation of the two out-scattering self-energies that appear in this paper. lead out
ðEÞ is self-energy due to leads,
q
ðEÞ is self-energy due to electron–phonon interaction, which is nonzero
which is nonzero only at the first and last device grid points. Phonon out
q
at all device grid points.
approximation, that the out-scattering self energy at
ðq; EÞ is [15]
out
Phononq;q ðEÞ
X h ¼
Dq ðnB h!phonon Þ þ 1 Gpq;q ðE h!phonon Þ
i
þ nB ð
h!phonon ÞGpq;q ðE þ h!phonon Þ :
term is similar except that it involves the right lead. In the
presence of electron–phonon interaction, the in-scattering
functions in
Phonon is nonzero at all grid points. As a result,
an electron can scatter from ðq0 ; E0 Þ to ðq0 ; EÞ and then
propagate to grid point ðq; EÞ via the term Gq;q0 ðEÞGyq0 ;q ðEÞ.
The expression for the electron density can be generalized
to include such terms
(94)
In the above equation, Gpq ðE h!phonon Þ and Gpq ðE þ h!phonon Þ
are the densities of unoccupied states at E h!phonon and
Eþ
h!phonon . So the first and second terms of (94) represent
out-scattering of electrons from E to E h!phonon (phonon
emission) and E þ h!phonon (phonon absorption), respectively. The out-scattering rate at grid point q is given by
nq ¼ 2
þ
(95)
where out
is the sum of all out-scattering self energies at
q
grid point q.
We now discuss how the in-scattering self-energy due
to electron–phonon scattering affects the expression for
electron density. The electron density at grid point q in the
phase-coherent case [(67) or (72)] is the sum of two terms
nq ¼ 2
Z
Z
h
qout ðEÞ
¼ out
q;q ðEÞ
dE h
y
Gq;1 ðEÞin
L ðEÞðG Þ1;q ðEÞ
2
i
y
þ Gq;n ðEÞin
ðEÞðG
Þ
ðEÞ
:
R
n;q
"
dE
y
Gq;1 ðEÞin
L ðEÞG1;q ðEÞ
2
y
þ Gq;n ðEÞin
R ðEÞGn;q ðEÞ
¼2
Out-scattering rate at grid point q :
Z
¼2
Z
X
#
y
Gq;q0 ðEÞin
q0 ;Phonon ðEÞGq0 ;q ðEÞ
q0
dE y
GðEÞin
lead ðEÞG ðEÞ
2
y
þ GðEÞin
Phonon ðEÞG ðEÞ q;q
dE n
G ðEÞ
2 q;q
(97)
where the third term corresponds to propagation of
electrons from grid point q0 to q after a scattering event
at q0 , as shown in Fig. 10. The in-scattering self-energies
due to phonon scattering are given by (92). More
generally, Gn is given by
Gn ðEÞ ¼ GðEÞin ðEÞGy ðEÞ
n
in
y
½E H ðEÞG ðEÞ ¼ ðEÞG ðEÞ
(98)
(99)
(96)
The first term represents in-scattering of electrons from
the left lead in
L ðEÞ, which is propagated to grid point q via
the term Gq;1 ðEÞGy1;q ðEÞ. The interpretation of the second
where in is the sum of self-energies due to leads and
electron–phonon interaction. The reader can compare the
above two equations to (73) and (83), which are valid in
the phase-coherent limit.
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
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Anantram et al.: Modeling of Nanoscale Devices
Fig. 10. Contributions to electron density from leads and electron–phonon scattering. The three terms of (97) are shown.
The density of unoccupied states can be written in a
manner identical to (97) as
hq ¼ 2
Z
"
dE
y
Gq;1 ðEÞout
L ðEÞG1;q ðEÞ
2
y
þ Gq;n ðEÞout
R ðEÞGn;q ðEÞ
þ
¼2
¼2
Z
Z
X
VI. NONEQUILIBRIUM GREEN’S
FUNCT I ON E QUAT I ONS FOR L AYE RED
STRUCTURES
#
y
Gq;q0 ðEÞout
q0 ;Phonon ðEÞGq0 ;q ðEÞ
q0
dE y
GðEÞout
lead ðEÞG ðEÞ
2
y
þ GðEÞout
Phonon ðEÞG ðEÞ q;q
dE p
G ðEÞ:
2 q;q
(100)
More generally, the Gp matrix is given by
Gp ðEÞ ¼ GðEÞout ðEÞGy ðEÞ
p
out
½E H ðEÞG ðEÞ ¼ y
ðEÞG ðEÞ
(101)
(102)
where out is the sum of self-energies due to leads,
electron–phonon interaction, and all other processes.
Note that, in general, Gn and Gp are full matrices, the
diagonal elements of which correspond to density of occupied
and unoccupied states, respectively, and the first off-diagonal
elements of Gn and Gp correspond to the current density.
The Green’s function G in the device region is obtained
by solving
½E H lead ðEÞ Phonon ðEÞG ¼ I
1526
which is similar to (60) for the phase-coherent case,
except for the additional self-energy due to phonon
scattering.
(103)
The previous section dealt with a simple one-dimensional
Hamiltonian. In this section, we will present the NEGF
equations for a family of more realistic structures called
layered structures. A layer can be considered to be a
generalization of a single grid point/orbital (Fig. 7) to a set
of grid points/orbitals per layer. Note that we will use Bgrid
points[ to represent both orbitals and the conventional
grid points that follow from discretization of a differential
equation in a real space grid. For example, consider the
structure that consists of two grid points per layer labeled
by a and b, as shown in Fig. 11(a). For this structure, the
form of the Hamiltonian remains the same as in (36)
except that q and tq;qþ1 become (2 2) matrices
!
!
ab
aq ab
taa
q
q;qþ1 tq;qþ1
and
, which represents the
bb
tba
ba
bq
q;qþ1 tq;qþ1
q
Hamiltonian element of layer q and the coupling between
layers q and qþ1, respectively. aq and bq are the diagonal
elements of the Hamiltonian at grid points a and b,
respectively, in layer q. tijq;p is the Hamiltonian element
representing interaction between grid point i in layer q and
a
grid point j in layer p, where i; j 2 a; b. q becomes qb ,
q
which is a (2 1) vector representing the wavefunction
in layer q. Its components aq and bq are the wave
functions at grid points a and b in layer q, respectively.
For the case in Fig. 11(a), the equivalent of Schrodinger’s
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
Fig. 11. (a) and (b) Scheme of simulation domains in the layered structure: device, left and right leads. (c) Examples of the layered structures:
carbon nanotube, DNA molecule, MOSFET.
equation for the single grid points per layer in (36)
becomes
taa
q;q1
tab
q;q1
tba
q;q1
tbb
q;q1
!
!
aq
þ
ba
b
0 E
q
q1
!
!
a
aa
ab
tq;qþ1 tq;qþ1
q
ba
b
tq;qþ1 tbb
q
q;qþ1
a
q1
E
0
ab
q
!!
bq
!
a
qþ1
b
qþ1
¼ 0:
Now, if we consider a structure with five grid points per
layer as shown in Fig. 11(b), the (2 2) matrices in the
above equation become (5 5) matrices.
Three examples of layered structures are shown in
Fig. 11(c). In each of these structures, a layer consists of the
orbitals/grid points between the dashed lines in Fig. 11. A
common approximation used to describe the Hamiltonian
of such layered structures consists of including interaction
only between nearest neighbor layers. That is, each layer q
interacts only with itself and its nearest neighbor layers
q 1 and q þ 1. Then, the single particle Schrodinger’s
equation of the layered system becomes
Tq;q1 q1 þ ðEI Hq Þq Tq;qþ1 qþ1 ¼ 0
where the size of Hq is equal to Nq and the size of Tq;qþ1 is
equal to ðNq Nqþ1 Þ, where Nq ðNqþ1 Þ is the number of
grid points in layer qðq þ 1Þ. Iq is an identity matrix of
dimension Nq . The Hamiltonian of the layered structure is
a block tridiagonal matrix, where diagonal blocks Hq
represent the Hamiltonian of layer q and off-diagonal
blocks Tq;qþ1 represent interaction between layers q and
q þ 1, can be written as shown in (104) at the bottom of
the next page, where we have made use of the fact that
y
Tiþ1;i ¼ Ti;iþ1
because the Hamiltonian is Hermitian. The
other elements of the Hamiltonian matrix are zero.
In the previous section, we derived the equations for the
Green’s function in the device by including the influence of
leads as self-energies rather rigorously within the single
particle picture. The self-energies due to electron–phonon
interaction, however, were introduced as an afterthought,
with the lead self-energies uninfluenced by electron–phonon
interaction [see (52) and (53)]. This is not correct, however,
because the electron–phonon interaction is present in the D,
L, and R regions. The Green’s function equation for the
entire device and leads in the presence of electron–phonon
scattering is given by [16]
½EI H Phonon G ¼ I
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
(105)
1527
Anantram et al.: Modeling of Nanoscale Devices
where Phonon is the self-energy due to electron–phonon
scattering. Partitioning of the device into the left lead (L), device
(D), and right lead (R) is a mathematical construct that is
motivated by the physics of the device. We partition the layered
structure into L, D, and R regions as shown in Fig. 11. The device
corresponds to the region where we solve for the nonequilibrium electron density. The leads are the highly conducting
regions connected to the nanodevice. While the device region,
where we seek the nonequilibrium density, consists of only n
layers, the matrix equation corresponding to (105) is infinite
dimensional due to the semi-infinite leads. We will now show
how the influence of the semi-infinite leads can be folded into
the device region. In a manner akin to the previous section, the
influence of the semi-infinite leads is to affect layers 1 and n of
the device region. An important difference is that the derivation
here includes electron–phonon scattering and does not assume
a flat potential in the semi-infinite leads. We first define
A0 ¼ ½EI H Phonon :
A0LL A0LD
B
A0 G ¼I !@ A0DL A0DD
O A0RD
0
1
I O O
B
C
¼@ O I O A
O O I
1
GLL GLD GLR
CB
C
A0DR A@ GDL GDD GDR A
A0RR
GRL GRD GRR
O
10
0
GLD ¼ A01
LL ALD GDD
0
GRD ¼ A01
RR ARD GDD
(112)
(113)
A0DL GLD þ A0DD GDD þ A0DR GRD ¼ I:
(114)
Substituting (112) and (113) in (114), we obtain a matrix
equation with dimension corresponding to total number of grid
points/orbitals in the n layers of the device
0
0
0
01 0
ADD A0DL A01
LL ALD ADR ARR ARD GDD ¼ I:
(106)
Noting that the Hamiltonian of the device can be partitioned
into the sub-Hamiltonians of the D, L, and R regions and
coupling between them, and noting that the Hamiltonian terms
coupling L and R are zero, (105) can be written as
0
where we have (108)–(111) as shown at the bottom of the
next page, where TLD ¼ Tl1;1 and TRD ¼ Tr1;n are the
coupling between the left and right leads and device,
respectively. Note that A0DL ¼ A0LD y , A0DR ¼ A0RD y , A0LD , and
A0DL (A0RD and A0DR ) are sparse matrices. Their only nonzero
entry represents coupling between the left (right) lead and
device. O represents zero matrices. From (107), we have
The second and third terms of (115) are self-energies due to
coupling of the device region to left and right leads, respectively.
The Green’s functions of the isolated semi-infinite
leads by definition are
A0LL gL ¼ I and A0RR gR ¼ I:
(107)
(115)
(116)
The surface Green’s function of the left and right leads are
the Green’s function elements corresponding to the edge
1
C
B C
B
C
B
C
B
y
C
B
Tl4;l3 Hl3 Tl3;l2
C
B
y
C
B
Tl3;l2 Hl2 Tl2;l1
C
B
C
B
y
C
B
Tl2;l1 Hl1 Tl1;1
C
B
C
B
y
Tl1;1 H1 T12
C
B
C
B
y
C
B
T12 H2 T2;3
C
B
C
B
C
B
C
H¼B
C
B
C
B
C
B
y
C
B
H
T
T
n1
n1;n
n2;n1
C
B
C
B
y
C
B
Tn1;n Hn Tn;r1
C
B
y
C
B
T
H
T
r1
r1;r2
n;r1
C
B
C
B
y
C
B
Tr1;r2 Hr2 Tr2;r3
C
B
y
C
B
T
H
T
r3
r3;r4
C
B
r2;r3
C
B
C
B
@
A
0
1528
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
(104Þ
Anantram et al.: Modeling of Nanoscale Devices
y
. Finally, defining matrix A to be (over the
A and TDR ¼ TRD
device layers 1 through n)
layers l1 and r1, respectively
01 gL l1;l1 ¼ A01
LL 1;1 and gR r1;r1 ¼ ARR 1;1 :
(117)
A ¼ EI H Phonon lead
(121)
(118) can be written as
Equation (115) can now be rewritten in a form very
similar to (60)
½EI H Phonon lead GDD ¼ I
(118)
where
lead1;1 ¼ TDL gL l1;l1 TLD ¼ L
(119)
leadn;n ¼ TDR gR r1;r1 TRD ¼ R :
(120)
All other elements of lead are zero. L and R are selfenergies due to the left and right leads, respectively, and
AG ¼ I:
(122)
Solving (122) gives us the Green’s function G over the
device layers 1 through n. The difference between and A
and A0 matrices above is that the former is defined only
over the device layer and has the effect of the semi-infinite
left (L) and right (R) leads included in it as self-energies.
The self-energies lead allowed us to derive (122) in the
device from (105), which was valid in the device and leads.
We stress that the elements of G in (122) are exactly equal
to the elements of G in the device region obtained from
(105). The main information needed to solve (118) is the
surface Green’s functions of gL and gR . We will discuss two
methods to obtain this surface Green’s functions for a
constant potential in the left and right leads. When the
potential does not vary, A0LL and A0RR are semi-infinite
0
1
B C
B
C
B
C
y
0
B
C
0
A
T
T
l3;l2
l3
l4;l3
ALL ¼ B
Ccorresponds to the left semi-infinite lead
B
C
y
0
B
Tl3;l2 Al2 Tl2;l1 C
@
A
y
Tl2;l1
A0l1
0
1
A0r1 Tr1;r2
B T y
C
A0r2 Tr2;r3
B
C
r1;r2
B
C
y
Ccorresponds to the right semi-infinite lead
0
A0RR ¼ B
Tr2;r3 Ar3 Tr3;r4
B
C
B
C
@
A
1
0 0
A1 T12
C
B
y
C
B T12
A02 T2;3
C
B
C
B
C
B
C
B
0
Ccorresponds to the device region
B
ADD ¼B
C
C
B
C
B
C
B
y
0
B
A
T
T
n1;n C
n1
n2;n1
A
@
y
0
Tn1;n
An
0
1
0
1
O
O O O
O O O TRD
B
C
B
C
O O OC
O C
B O
BO O O
B
C
B
C
0
B
A0LD ¼ B
O O OC
O C
B O
C and ARD ¼ B O O O
C
B
C
B
C
O O OA
O A
@ O
@O O O
TLD O O O
O O O O
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
(108)
(109)
(110)
(111)
1529
Anantram et al.: Modeling of Nanoscale Devices
periodic matrices with all diagonal/off-diagonal blocks
being equal
A0l1 ¼ A0l2 ¼ A0l3 ¼ . . . ¼ A0l
A0r1 ¼ A0r2 ¼ A0r3 ¼ . . . ¼ A0r
(123)
Tl1;l2 ¼ Tl2;l3 ¼ Tl3;l4 ¼ . . . ¼ Tl
Tr1;r2 ¼ Tr2;r3 ¼ Tr3;r4 ¼ . . . ¼ Tr :
(124)
D, and R; its dimension is infinite due to the semi-infinite
left and right leads. It can, however, be converted to a
finite dimensional matrix with dimension equal to the
number of grid points/orbitals corresponding to the n
device layers by calculating the self-energy due to the L
and R leads. In a manner identical to the derivation of
(118) for the Green’s function, it can be shown that the
leads can be folded into layers 1 and n of the device to yield
½EI H Phonon lead Gn ¼ in Gy
gL l1;l1 is obtained by solving the matrix quadratic equation
h
i
A0l Tly gL l1;l1 Tl gL l1;l1 ¼ I:
(130)
or
(125)
AGn ¼ in Gy
(131)
This equation can be solved iteratively by
h
i
hm1i
hmi
A0l Tly gL l1;l1 Tl gL l1;l1 ¼ I
(126)
where the superscript of gL represents the iteration
number. Note that the solution to (125) is analytic when
the dimension of Al is one. A second simpler solution to
obtain gL l1;l1 involves transforming to an eigenmode basis
using an unitary transformation (S), such that
S1 A0l S
¼ ALdiag
1
S Tl S ¼ TLdiag
and
where A has been defined in (121). Equation (131) gives the
electron density only in the device (D) regions. We stress
that the elements of Gn in (131) are exactly equal to the
elements of Gn in the device region obtained from (129).
The self-energy in defined over device layers 1
through n has contributions due to both electron–phonon
interaction and leads
(127)
where both ALdiag and TLdiag are diagonal matrices. The
surface Green’s function in this new basis is simply a
diagonal matrix, whose elements are obtained by solving
the scalar quadratic version of (125). The Green’s function
in the original basis (in which A0l is not diagonal) can be
obtained using the inverse unitary transformation.
1) Electron ðGn Þ and Hole ðGp Þ Green’s Function: The
electron density at any location in D, L, or R is equal to
(see the discussion of electron density in Section V-C)
G ð~
r;~
r; EÞ
:
2
The governing equation for G is
q ¼ 2; 3; 4; . . . ; n1:
(134)
in
L ðEÞ ¼ 2Im½L ðEÞfL ðEÞ ¼ L ðEÞfL ðEÞ
(135)
in
R ðEÞ ¼
(136)
2Im½R ðEÞfR ðEÞ ¼ R ðEÞfR ðEÞ
where
L ðEÞ ¼ 2Im½L ðEÞ
(137)
R ðEÞ ¼ 2Im½R ðEÞ:
(138)
(129)
fL and fR are the distribution functions in the left and
right leads, respectively (Fermi factors at equilibrium).
The self-energies L ðEÞ and R ðEÞ have been defined in
(119) and (120).
The diagonal elements of the hole correlation functions
Gp ðEÞ represents the density of unoccupied states
where Gy is the Hermitian conjugate Green’s function and
in
Phonon is the in-scattering self-energy due to phonon
scattering. Equation (129) gives the electron density in L,
1530
(133)
in
¼ in
Phononn;n þ R
¼ in
Phononq;q ; where
(128)
n
y
½EI H Phonon Gn ¼ in
Phonon G
(132)
in
n;n
in
q;q
in
The in-scattering self-energies due to the leads in
L and R
have forms very similar to (70) and (71) of Section V-C
n
nð~
r; EÞ ¼ 2
in
in
in
1;1 ¼ Phonon1;1 þ L
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
hð~
r; EÞ ¼ 2
Gp ð~
r;~
r; EÞ
:
2
(139)
Anantram et al.: Modeling of Nanoscale Devices
The governing equation for Gp ðEÞ is
y
½EI H Phonon Gp ¼ out
Phonon G
(140)
where Tr stands for trace. Equation (149) frequently
appears in the literature in other useful forms that are
derived below. Expanding both terms of (149) using (B10)
of Appendix I, we get
or
ADD Gp ¼ out Gy
out
out 1;1 ¼ out
Phonon1;1 þ L
(141)
(142)
out
out n;n ¼ out
Phononn;n þ R
(143)
out
out
i;i ¼ Phononi;i ;
where i ¼ 2; 3; 4; . . . n 1
ie
JL ¼ 2
h
Z
i
þ TLD Gn 1;1 ðEÞTDL gLy l1;l1 ðEÞ
h
TDL gLn l1;l1 ðEÞTLD Gy 1;1 ðEÞ
i
þ TDL gLy l1;l1 ðEÞTLD Gn 1;1 ðEÞ
(144)
where out
Phonon is the out-scattering self-energy due to
phonon scattering. The out-scattering self-energies due to the
out
leads out
L and R have forms very similar to (88) and (89)
Z
ie
dE
Tr
¼ 2
h
2
¼ R ð1 fR ðEÞÞ
(146)
Gn ¼ A1 in Gy ¼ Gin Gy
(147)
Gp ¼ A1 out Gy ¼ Gout Gy :
(148)
While these equations appear often in literature, we do not
suggest using them to compute the diagonal elements of Gn
and Gp of layered structures. This is because their use requires
knowledge of the entire G matrix when in is nonzero at all
grid points. Computation of the entire G amounts to inversion
of A. Matrix inversion is computationally expensive. The
diagonal elements of Gn and Gp of layered structures can be
computed more efficiently [23] without calculating the entire
G matrix using the algorithm developed in [23], a simplified
version of which is discussed in Appendix II.
2) Current Density: We will now present some expression for current density commonly used in literature. In
doing so, for brevity of notation we will drop the subscript
DD used in the above sections, and simply remember that
the Green’s functions represented by G and Gn are over the
device layers 1 through n. The current flowing between
layers q and q þ 1 is [which has a similar form to (85)]
Jq!qþ1
ie
¼ 2
h
(150)
G1;1 ðEÞGy 1;1 ðEÞ TDL gLn l1;l1 ðEÞTLD
(151)
(145)
Using the relationships
n
in
L ¼ TDL gL l1;l1 TLD
h
i
iL ¼ TDL gL l1;l1 gLy l1;l1 TLD :
where 1fL ðEÞ and 1fR ðEÞ are the probabilities of finding
an unoccupied state in the left and right lead at energy E.
Equations (131) and (141) for Gn and Gp can be written as
Z
Gn 1;1 ðEÞTDL
h
i gL l1;l1 ðEÞgLy l1;l1 ðEÞ TLD :
out
L ðEÞ ¼ 2Im½L ðEÞð1 fL ðEÞÞ
¼ L ð1 fL ðEÞÞ
out
ðEÞ
¼
2Im½R ðEÞð1 fR ðEÞÞ
R
dE h
Tr TLD G1;1 ðEÞTDL gLn l1;l1 ðEÞ
2
dE Tr Tq;qþ1 Gn qþ1;q ðEÞ
2
Tqþ1;q Gn q;qþ1 ðEÞ (149)
(152)
(153)
Equation (151) can be written as
e
JL ¼ 2
h
Z
dE Tr i G1;1 ðEÞ Gy 1;1 ðEÞ
2
n
in
L ðEÞ G 1;1 ðEÞL ðEÞ : (154)
Equations (149) and (154) are both general expression for
current density valid in the presence of electron–phonon
scattering in the device [17]. The advantage of using (149)
is that the current density can be calculated at every layer of
the device. This expression is useful in understanding how
the current density is energetically redistributed along the
length of the device as a result of scattering.
In the phase-coherent limit, where Phonon ¼ 0, we
expect to get the Landauer–Buttiker formula, which is a
~ L and ~R,
special case of (149) and (154). We define matrices which consist of n diagonal blocks corresponding to the n
device layers (dimension of A matrix) and with the following
nonzero elements:
~ L j ¼ L ;
1;1
~ R j ¼ R :
n;n
(155)
Now left-multiplying (122) by Gy and right-multiplying the
Hermitian conjugate of (122) by G, and subtracting the
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Anantram et al.: Modeling of Nanoscale Devices
resulting two equations, we have
G Gy ¼ Gy ð y ÞG
The total transmission at energy E is identified from (159) to be
where is the total self-energy due to phonons and the leads.
The Hermitian conjugate Green’s functions and self-energies
are Gy and y . In the absence of phonon scattering, the selfenergies only have components due to the leads and so (156)
can be written as
~L þ ~ R ÞG
i½G Gy ¼ Gy ð
(157)
where (137) and (138) have been used. It also follows from
(135), (136), and (147) that
~ L fL þ ~ R fR ÞGy :
Gn ¼ Gð
(158)
Now using (157) and (158) in (154), the current in the phasecoherent limit is
e
JL ¼ 2
h
Z
dE
TðEÞ½fL ðEÞ fR ðEÞ:
2
~ R ðEÞGy ðEÞ :
~ L ðEÞGðEÞ
TðEÞ ¼ Tr (156)
(159)
Note that to compute the total transmission using (160), only
the elements of G connecting layers 1 and n are required because
~ L and ~ R are nonzero only in layers 1 and n, respectively.
A. Crib Sheet
The algorithmic flow in modeling nanodevices using the
nonequilibrium Green’s function consists of the following
steps (Fig. 12). We first find a guess for the electrostatic
potential Vð~
r Þ and calculate the self energies due to the leads
[(175)–(185)]. The self-energies due to electron–phonon
scattering are set to zero. The nonequilibrium Green’s
function equations for G, Gn , and Gp [(171)–(174)] are then
solved. Following this, the self-energies due to electron–
phonon scattering and leads [(175)–(185)] are calculated. As
the equations governing the Green’s functions depend on the
self-energies, we iteratively solve for the Green’s function
and self-energies, as indicated by the inner loop of Fig. 12.
Then, the electron density (diagonal elements of Gn ) is used
in Poisson’s equation to obtain a new potential profile. We
use this updated electrostatic potential profile as an input to
Fig. 12. Flowchart of a typical simulation involved in modeling of a nanodevice. (Adapted from [1].)
1532
(160)
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Anantram et al.: Modeling of Nanoscale Devices
solve for updated nonequilibrium Green’s functions and
continue the above process iteratively until convergence is
achieved (outer loop of Fig. 12). A number of equations that
are repeatedly used in nanodevice modeling are listed below.
Current density flowing from the left lead into layer 1 of
device (valid only in the phase coherent limit) is
JL ¼
1) Physical Quantities:
Scattering Rate
h
¼ ðEÞ ¼ 2 Im½ðEÞ
ðEÞ
h
¼ in;out ðEÞ
in;out
ðEÞ
2e
h
Z
dE
TðEÞ½fL ðEÞ fR ðEÞ
2
(169)
where the total transmission from the left to right lead at
energy E is given by
(161)
~ L ðEÞGðEÞ
~ R ðEÞGy ðEÞ :
TðEÞ ¼ Tr (162)
(170)
Only elements of G connecting layers 1 and n are necessary.
Density of States (DOS) at ð~
r; EÞ. Use recursive algorithm
to calculate DOS (Do not invert A).
1
r;~
r; EÞÞ
Nð~
r; EÞ ¼ ImðGð~
2) Equations Solved:
Green’s Function:
½EI H GðEÞ ¼ I ! AG ¼ I
Hermitian conjugate Green’s Function:
Gy ðEÞ½EI H y ¼ I
electron correlation function:
½EI H Gn ðEÞ ¼ in ðEÞGy ðEÞ
! AGn ¼ in Gy
hole correlation function:
½EI H Gp ðEÞ ¼ out ðEÞGy ðEÞ
! AGp ¼ out Gy
(163Þ
Electron/Occupied density at ~
r. Use recursive algorithm to
calculate n (Do not use Gn ¼ Gin Gy ).
nð~
rÞ ¼ 2
Z
dE n
G ð~
r;~
r; EÞ
2
(164)
Hole/Unoccupied Density at ~
r. Use recursive algorithm to
calculate h (Do not use Gp ¼ Gout Gy ).
hð~
rÞ ¼ 2
Z
dE p
G ð~
r;~
r; EÞ
2
ðEÞ ¼ lead ðEÞ þ Phonon ðEÞ;
where 2 in,out
(165)
Jq!qþ1 ¼
ie
2
h
Z
i
dE h
Tr Tq;qþ1 Gnqþ1;q ðEÞ Tqþ1;q Gnq;qþ1 ðEÞ
2
(166)
Current density flowing from the left lead into layer 1 of
device (valid with scattering in device)
Z
i
dE n h
Tr i G1;1 ðEÞ Gy1;1 ðEÞ in
L ðEÞ
2
o
Gn1;1 ðEÞL ðEÞ
Z
e
dE n p
Tr G1;1 ðEÞin
¼ 2
L ðEÞ
h
2
o
Gn1;1 ðEÞout
ðEÞ
:
L
e
JL ¼ 2
h
(167)
(168)
(172)
(173)
(174)
(175)
lead1;1 ðEÞ ¼ L ðEÞ
(176)
leadn;n ðEÞ ¼ R ðEÞ
(177)
leadi;i
Current density flowing between layers q and q þ 1
(valid with scattering in device):
(171)
¼ 0 8i 6¼ 1; n
(178)
ðEÞ ¼ 2Im½ðEÞ
(179)
L ðEÞ ¼ 2Im½L ðEÞ
R ðEÞ ¼ 2Im½R ðEÞ
(180)
(181)
in
L ðEÞ ¼ L ðEÞfL ðEÞ
(182)
and
in
R ðEÞ
out
L ðEÞ
out
R ðEÞ
¼ R ðEÞfR ðEÞ
(183)
¼ L ½1 fL ðEÞ
¼ R ½1 fR ðEÞ:
(184)
(185)
The diagonal and nearest neighbor off-diagonal elements
of G and Gn are computed repeatedly as they correspond to
physical quantities such as the density of states, electron
density, and current. Nonlocal scattering mechanisms,
which require calculation of further off-diagonal elements,
are not discussed here.
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Anantram et al.: Modeling of Nanoscale Devices
3) Useful Relationships:
i½G Gy ¼ Gp þ Gn / DOS
y
out
i½ ¼ y
(186)
in
þ ¼
y
(187)
y
G G ¼ G½ G
y
(188)
y
¼ G ½ G
y
(189)
y
¼ iGG ¼ iG G
Gny ¼ Gn
G
py
where ðmbx ; mby ; mbz Þ are the ðx; y; zÞ components of the
electron effective mass in valley b and VðrÞ is the potential
energy. The equations for the Green’s function ðGÞ and
electron and hole correlation functions (Gn and Gp ) are
p
¼G :
½E Hb1 ð~
r1 ÞGb1 ;b2 ð~
r1 ;~
r2 ; EÞ
Z
d~
r b1 ;b0 ð~
r1 ;~
r; EÞGb0 ;b2 ð~
r;~
r2 ; EÞ ¼ b1 ;b2 ð~
r1 ~
r2 Þ
(194)
(190)
(191)
(192)
and
nðpÞ
½E Hb1 ð~
r1 ÞGb1 ;b2 ð~
r1 ;~
r2 ; EÞ
Z
nðpÞ
d~
r b1 ;b0 ð~
r1 ;~
r; EÞGb0 ;b2 ð~
r;~
r2 ; EÞ
Z
inðoutÞ
r1 ;~
r; EÞGyb0 ;b2 ð~
r;~
r2 ; EÞ:
¼ d~
r b1 ;b0 ð~
VI I. A PP L ICAT I ON TO A B AL L IS TI C
NANOTRANSISTOR
Quantum mechanics is playing an increasingly important
role in modeling transistors with channel lengths in the
10 nm regime for several reasons.
i) Tunneling from gate to channel and source to
drain determine the off current.
ii) Ballistic flow of electrons in the channel is
important as the channel length becomes comparable to the electron mean free path.
iii) Classically, the electron distribution in the inversion layer is a sheet charge at the Si-SiO2 interface.
Quantum mechanically, the inversion layer charge is
distributed over a few nanometers perpendicular to the SiSiO2 interface due to quantum confinement. Methods based
on the drift-diffusion and Boltzmann equations do not a priori
capture the quantum mechanical features mentioned above.
In this section, we will first discuss the NEGF equations
involved in the two-dimensional modeling of nanotransistors
within the effective mass framework [19], [23] and then
compare the quantum mechanical and semiclassical results to
point out the importance of the quantum (NEGF) formulation. The related equations and their discretized matrix forms
that we solve are discussed in Section VII-A, while the
application of the quantum mechanical method to illustrate i)
the role of the polysilicon gate depletion, ii) the slopes of the
drain current versus gate voltage ðId –Vg Þ, and iii) the transmission function TðEÞ are discussed in Section VII-B.
A. Related Equations and Discretization
The schematic of the cross-section of the simulated
nanoscale MOSFET is shown in Fig. 13. We consider Nb
independent valleys for electrons within the effective mass
approximation. The Hamiltonian of valley b is
!
" h2 d 1 d
d 1 d
Hb ð~
rÞ ¼ þ
dy mby dy
2 dx mbx dx
#
d 1 d
þ
þ eVð~
rÞ (193)
dz mbz dz
1534
(195)
The coordinate in (194) and (195) spans only the device.
The influence of the semi-infinite source (S), drain (D),
and polysilicon gate (G) leads and the electron–phonon
interaction are included via self-energy terms b1 ;b0 and
inðoutÞ
b1 ;b0 , as discussed in Section VI. The lead selfenergies are diagonal in the band index b1 ;b2 ;C ¼ b1 ;C b1 ;b2
(C represents contacts ¼ leads).
The electrostatic potential VðrÞ varies in the ðx; yÞ
plane of Fig. 13, and the system is translationally
invariant along the z-axis. So, any quantity Qð~
r1 ;~
r2 ; EÞ
depends only on the difference coordinate z1 z2 . Using
the relation
r2 ; EÞ ¼
Qð~
r1 ;~
Z
dkz ikz ðz1 z2 Þ
e
Qðx1 ; y1 ; x2 ; y2 ; kz ; EÞ (196)
2
the equations for G and GnðpÞ simplify to
h2 k2z
E
Hb ð~
r1 Þ Gb ð~
r1 ;~
r2 ; kz ; EÞ
2mz
Z
d~
r b ð~
r1 ;~
r; kz ; EÞGb ð~
r;~
r2 ; kz ; EÞ ¼ ð~
r1 ~
r2 Þ (197)
and
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
h2 k2z
nðpÞ
E
Hb ð~
r1 Þ Gb ð~
r1 ;~
r2 ; kz ; EÞ
2mz
Z
nðpÞ
d~
r b ð~
r1 ;~
r; kz ; EÞGb ð~
r;~
r2 ; kz ; EÞ
Z
inðoutÞ
ð~
r1 ;~
r; kz ; EÞGyb ð~
r;~
r2 ; kz ; EÞ
¼ d~
r b
(198)
Anantram et al.: Modeling of Nanoscale Devices
Fig. 13. Schematic of the cross-section of the nanoscale MOSFET simulated. The equations are solved in a two-dimensional nonuniform
spatial grid, with semi-infinite boundaries as shown. Each column q corresponds to the diagonal blocks of the Green’s function equations.
(Adapted from [23].)
where Gb ¼ Gb;b0 b;b0 and b ¼ b;b0 b;b0 , and ~
r ! ðx; yÞ
for the remainder of this section.
Equations (197) and (198) can be written in matrix
form as
AG ¼ AG
nðpÞ
¼
Table 1 List of Abbreviations: Length Scales
(199)
inðoutÞ y
G:
(200)
The self-energies due to S, D, and G are nonzero only along
the lines y ¼ Ly =2, y ¼ þLy =2, and x ¼ ðLP þ tox Þ of
Fig. 13. Definitions of symbols for length variables are given in
Table 1. The A matrix is ordered such that all grid points at a ycoordinate (layer) correspond to a diagonal block of
dimension Nx and there are N such blocks. In the notation
adopted, Aq1 ;q2 ði; i0 Þ is the entry corresponding to grid points
ðxi ; yq1 Þ and ðxi0 ; yq2 Þ. The index q refers to the layers in
Section VI and corresponds to the y-direction in Fig. 13. The
nonzero elements of the diagonal blocks of the A matrix are
Aq;q ði; iÞ ¼ E0 Vi;q Tq;q ði þ 1; iÞ Tq;q ði 1; iÞ
Tqþ1;q ði; iÞ Tq1;q ði; iÞ S ðxi ; xi Þq;1
D ðxi ; xi Þq;N G ðyq ; yq Þi;1
Aq;q ði 1; iÞ ¼ Tq;q ði 1; iÞ S ðxi1 ; xi Þq;1
(201)
D ðxi1 ; xi Þq;N
(202)
0
Aq;q ði; i Þ ¼ S ðxi ; x Þq;1 D ðxi ; x Þq;N ;
for i0 6¼ i; i 1
i0
i0
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(203)
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Anantram et al.: Modeling of Nanoscale Devices
where E0 ¼ E h2 k2z =2mz and Vi;q ¼ Vðxi ; yq Þ. The offdiagonal blocks are
Aq1;q ði; iÞ ¼ Tq1;q ði; iÞ G ðyq ; yq1 Þi;1
Aq;q0 ði; i0 Þ ¼ 0; for q0 ¼
6 q; q 1
(204)
and the nonzero elements of the T matrix are
h2
2
1
x
2m xiþ1 xi1 jxi1 xi j
h2
2
1
Tq1;q ði; iÞ ¼
y
2m yqþ1 yq1 jyq1 yq j
Tq;q ði 1; iÞ ¼
(205)
(206)
where mx ¼ ðmi1;q þ mi;q Þ=2 and my ¼ ðmi;q1 þ mi;q Þ=2.
Nonzero elements of G ðyq ; yq0 Þ, where q0 6¼ q, q 1 are
neglected to ensure that A is block tridiagonal. The algorithm
to calculate G and Gn (Appendix II) relies on the block
tridiagonal form of A. The appearing in (199) corresponds to
the delta function in (197). is a diagonal matrix whose
elements are given by
q;q ði; iÞ ¼
4
:
ðxiþ1 xi1 Þðyqþ1 yq1 Þ
(207)
B. Results
We now discuss the aspects of the quantum mechanical
transport in a two-dimensional ballistic nanotransistor.
Fig. 14. Plot of drain current versus gate voltage from the quantum
mechanical and drift-diffusion (MEDICI) calculations at Vd ¼ 1 V.
At small gate voltages, the drain current from MEDICI is comparable
to ‘‘flat band in poly.’’ The drain current from ‘‘quantum treatment
of poly’’ is, however, significantly different at all gate voltages.
(From [23].)
1536
Fig. 15. Potential profile at the y ¼ 0 slice of MIT25, calculated using
quantum and drift-diffusion methods by assuming flat band in the
polysilicon gate. (From [23].)
The BMIT well-tempered 25 nm[ device,1 which is
referred to here as MIT25, is considered for the purpose
of discussions. The nþ doping is 2 1020 cm3 in both
the source and drain regions while the nþ doping is
5 1020 cm3 in the polysilicon region. MIT25 has a
gate width of 50 nm and oxide thickness of 1.5 nm. The
effective channel length, defined here as the distance
between the source and drain positions where the doping
falls down to 2 1019 cm3 , is around 25 nm. In all
calculations, we assume an isotropic effective mass for
electrons. BQuantum treatment of poly[ refers to
computing the electrostatic potential in the polysilicon
region by setting the Poisson boundary condition for the
electrostatic potential deep inside the polysilicon region
and computing the electron density in the polysilicon
region quantum mechanically, while Bflat band in poly[
refers to neglecting the potential drop in the polysilicon
region by setting the boundary condition for the electrostatic profile at the oxide-poly interface (at y ¼ tox).
We first compare the current ðId Þ versus voltage ðVg Þ
characteristics from our quantum and drift-diffusion
(using MEDICI) simulations, as shown in Fig. 14. In the
quantum case, there are higher off-current, higher
threshold voltage shift, smaller subthreshold slope, and
much higher on-current. The change in the threshold
voltage results directly from the very different boundary
for potential and the quantum treatment of electrons in
the polysilicon gate.
We now compare the conduction band profiles for the
two cases in Fig. 15. At polysilicon near the oxide-poly interface, the quantum band bending is opposite to that for the
drift-diffusion case. For the quantum case, the conduction
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
1
http://www-mtl.mit.edu/researchgroups/Well/.
Anantram et al.: Modeling of Nanoscale Devices
Fig. 16. Drain current versus gate voltage for Vd ¼ 1 V. Quantum
mechanical treatment of the polysilicon gate results in much higher
current (solid line). The triangles correspond to the Id ðVg Þ calculated
using a flat band in the polysilicon region shifted by the equilibrium
built-in potential of 130 meV in the polysilicon region. (From [23].)
band is lower by approximately 130 meV. The physical
reason for the difference is that, although classically
electron density close to the interface is very high, quantum
mechanically the electron density there is very tiny. This is
because the electron wave function is tiny at the oxide
interface because of the large barrier due to the oxide. As
the density near the interface is much smaller than the
uniform background doping density, the conduction band
bends in a direction opposite to that computed classically.
In Fig. 16, the Id – Vg characteristics is plotted both for
Bquantum treatment of poly[ and Bflat band in poly[ cases.
Also shown is the Id – Vg characteristic in the Bflat band in
poly[ case shifted by the 1-D equilibrium built-in potential
[23]. The gate voltage shift is approximately equal to the
band bending in the polysilicon gate. The 1-D built-in
potential shifted Bflat band in poly[ band is close to the
Bquantum treatment of poly[ band, but the difference
increases at higher gate voltages.
Fig. 17 plots the heights of the first quantum resonant
level ðEr1 Þ and the classical source injection barrier ðEb Þ
versus the gate voltage (a). Also plotted in the same figure
is the narrowing of the triangular well in the channel with
increase in the gate voltage (b). We see that, with
increase in Vg , Er1 decreases more slowly compared to
both Eb . The slower variation of Er1 arises due to quantum
confinement in the triangular well in the channel that
becomes progressively narrower with increase in gate
voltage as shown in the right part of Fig. 16. This change
in confinement is not an issue in the classical case.
Because of the slow variation of Er1 with the gate voltage
the subthreshold slope d½logðId Þ=dVg is smaller in the
quantum case compared to the classical case or driftdiffusion case (Fig. 14). We note here that the
subthreshold current resulting from the simple intuitive
expression
Er1
I ¼ Iq0 e kT
(208)
matches the quantum result quite accurately. Here Iq0
is a prefactor chosen to match the current at Vg ¼ 0.
Fig. 17. (a) Location of the first resonant level Er1 and the classical source injection barrier Eb (classical) versus gate voltage. Note that Er1
decreases slower than Eb (classical) with gate voltage due to narrowing of channel potential well. (b) Narrowing of the triangular well in the
channel with increase in gate bias. Eb (classical) is the bottom of the triangular well, and the resonant level is shown by the horizontal line.
(Adapted from [23].)
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Anantram et al.: Modeling of Nanoscale Devices
states as shown in the inset of Fig. 18. Fourthly, the
transition from one step to the next higher step develops
over an energy window of about 50 meV. Interestingly,
in the case of MIT25, the current is predominantly
carried at energies where the transmission is not an
integer.
VIII. APPLICATION T O
NANOTRANSISTORS WITH
ELECTRON–PHONON SCATTERI NG
Fig. 18. Transmission ðþÞ and density of states (solid) versus energy at
a spatial location close to the source injection barrier, at Vg ¼ 0 V and
Vd ¼ 1 V. The peaks in the density of states represent the resonant
levels in the channel. (Inset) The density of states at three different
y-locations and total transmission ðþÞ. The points y ¼ 7 and 0 nm are
to the left and right of the location where the source injection barrier is
largest (close to y ¼ 4 nm). (From [23].)
The higher resonant levels do not carry appreciable
currents.
We will now address the value of total transmission in a
ballistic MOSFET. The transmission is related to drain
current ðId Þ by (159)
Id ¼
2e
h
Z
dE TðEÞ½fS ðEÞ fD ðEÞ
(209)
where T is the total transmission from source to drain. fS
and fD are the Fermi factors in the source and drain,
respectively, and the factor of two accounts for spin. As
an electron transits from source to drain, the main
factors that determine the transmission probability are
the tunneling and the scattering in the two-dimensional
potential profile. The quantum mechanically computed
transmission versus energy is shown in Fig. 18. Four
points can be noticed from the transmission curve. First,
the transmission increases in a step-wise manner, with
the integer values at the plateaus equal to the number of
conducting modes in the channel. Secondly, the steps
turn on at an energy determined by the effective
Bsubband dependent[ source injection barrier (which
depends on Er1 ), that is, the maximum subband energy
between the source and drain due to quantization
perpendicular to the gate plane (x-direction of Fig. 13).
Thirdly, the total transmission assumes integer values at
an energy slightly above the maximum in 2-D density of
1538
The channel, scattering, and screening lengths become
comparable in transistors with diminishing channel
lengths and the ballistic transport becomes important
(Section VII). However, carrier transport is not fully
ballistic. Realistic nanodevice modeling will involve
phase-breaking scattering such that transport is between
the ballistic and diffusive limits. In this regime, in
contrast to long channel devices, carriers are not
thermally relaxed in the drain-end of the transistor and
are reflected towards the source-end. This reflection of
the hot carriers should be explicitly included in the
models to compute the drive current. It is in this
intermediate regime that the NEGF (with Poisson)
method has an advantage over solving the Schrodinger
and Poisson equations self-consistently.
In this section, we will first discuss the NEGF
equations involved in the two-dimensional modeling of
a dual-gate MOSFET (DGMOSFET) with electron–
phonon scattering and then illustrate the effect of scattering on the MOSFET characteristics [24]. The related
equations, associated approximations and the discretized
equations that we solve, are discussed in Section VIII-A,
while the application of the NEGF method to demonstrate the role of scattering on transport is discussed in
Section VIII-B.
A. Related Equations, Approximations,
and Discretization
Fig. 19 shows the schematic of the simulated device. If
we consider only the gate-to-gate direction, then we have a
1-D potential well sandwiched between the two oxides so
that the quantized energy levels in the channel (well) are
approximately given by
En ¼
n2 2 h2
2
2mx Tch
(210)
where mx is the electron effective mass along the
x-direction and Tch is the channel thickness, as shown in
Fig. 19. When Tch is small, only a few (usually less than
four) subbands determine the current–voltage characteristics. In such a case, the mode space approach [27]
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
gives results comparable to a full 2-D simulation. The mode
space approach consists of solving a 1-D Schrodinger
equation along the x-direction at each y-position (layer). It
is computationally very efficient as it avoids solving a full 2-D
Schrodinger equation. The lowest three quantized energy
levels for a device with Tch ¼ 1:5 nm are 173, 691, and
891 meV above the bulk conduction band. The Fermi
energy at the lead doping of 1 1020 cm3 is 60 meV
above the bulk conduction band. As a result, electrons are
injected only into the first subband from the source-end at
the operating voltage of Vd ¼ Vg ¼ 0:6 V.
The three-dimensional effective mass Hamiltonian is
the same as (193). Noting that if the z-direction is
infinite, the wave function can be expanded as
ðx; y; zÞ ¼ eikz z ðx; yÞ. Schrodinger’s equation is then
"
E
h2 k2z
2mbz
2
!
In the first step of the mode space approach, we solve
the following x-directed 1-D Schrodinger equation at each
y-position (layer):
h2 d 1 d
þ Vðx; yÞ n ðx; yÞ
2 dx mbx dx
¼ En ðyÞn ðx; yÞ
"
h2 k2
E nz 2mz
(212)
where En ðyÞ is the subband energy and n ðx; yÞ, the 1-D
x-directed wavefunction, both at a certain y. Here
n ¼ ; b, where is the quantum number due to
h2 d 1 d
2 dy mny dy
!#
!
þ En ðyÞ
Gn ðy; y0 ; kz ; EÞ
Z
dy1 n ðy; y1 ; kz ; EÞGn ðy1 ; y0 ; kz ; EÞ
¼ ðy y0 Þ;
"
h d 1 d
2 dy mby dy
#
h2 d 1 d
þ Vðx; yÞ ðx; yÞ ¼ 0: (211)
2 dx mbx dx
quantization in the x-direction and b ¼ 1; 2; and 3
are the valley indexes.
Then, the Green’s function equations for G, Gn and Gp
are solved for each subband n
E
h2 k2z
2mnz
and
(213)
h2 d 1 d
2 dy mny dy
!
!#
þ En ðyÞ
GnnðpÞ ðy; y0 ; kz ; EÞ
Z
0
dy1 n ðy; y1 ; kz ; EÞGnðpÞ
n ðy1 ; y ; kz ; EÞ
¼
Z
dy1 inðoutÞ
ðy; y1 ; kz ; EÞ
n
Gyn ðy1 ; y0 ; kz ; EÞ:
(214)
mny and mnz are the effective masses of silicon in the y- and
z-directions that give rise to subband index n. En ðyÞ is
effectively an electrostatic potential for electrons in
subband n. Note that in (213) and (214), the subscript n
refers to the subband index while the superscript n refers
to the type of Green’s function.
The self-energies can be written as
n ¼ n;C þ n;Phonon
n;Phonon
Fig. 19. Schematic of a DGMOSFET. Ex-s and Ex-d are the extension
regions and the hatched region is the channel. The white region
between the source/drain/channel and gate is the oxide. The device
dimension normal to the page is infinite in extent. (From [24].)
¼ n;el
þ
n;inel
(215)
(216)
where 2 (empty), out, in. n;C is the self-energy due to
the leads (contacts). The phonon self-energy n;Phonon
consists of two terms: n;el due to elastic and n;inel due to
inelastic scattering. Only the lowest three subbands and
the electron–phonon scattering among them are considered. All other subbands and the scattering associated with
them are neglected because the population of the higher
subbands is negligible. The self-energy due to leads is
nonzero only at the first (source) and last (drain) grid
points.
Assuming isotropic scattering and a phonon reservoir in equilibrium, and using the self-consistent Born
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
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Anantram et al.: Modeling of Nanoscale Devices
approximation, the self-energies due to electron–phonon
scattering at grid point yi are [15]
el;n ðyi ; EÞ
pffiffiffiffiffiffi0ffi Z
X
mn
1
el
Dn;n0 pzffiffiffi dEz pffiffiffiffiffi Gn0 ðyi ; Ez ; EÞ
¼
Ez
h
2
n0
(217)
in
inel;n ðyi ; EÞ
pffiffiffiffiffiffi0ffi
X i;
mn
Dn;n0 pzffiffiffi
¼
h 2
n0 ;
Z
1 dEz pffiffiffiffiffi nB ð
h! ÞGnn0 ðyi ; Ez ; E h! Þ
Ez
þ nB ð
h! Þþ1 Gnn0 ðyi ; Ez ; Eþ
h ! Þ
energetic redistribution of carriers are included but the
real part is set to zero. While this approximation has
worked well for transistors, it should be used with
caution in other situations [24].
In the numerical solution, N uniformly spaced grid
points in the y-direction with the grid spacing equal to
y are considered. The discretized form of (213) and
(214) is then
Ai;i Gn ðyi ; yi0 ; kz ; EÞ þ Ai;iþ1 Gn yiþ1 ; y0i ; kz ; E
þ Ai;i1 Gn yi1 ; y0i ; kz ; E
i;i0
; and
¼
y
Ai;i Gn yi ; y0i ; kz ; E þ Ai;iþ1 Gn yiþ1 ; y0i ; kz ; E
þ Ai;i1 Gn yi1 ; y0i ; kz ; E
¼ n ðyi ; EÞGyn yi ; y0i ; kz ; E
(218)
and
out
inel;n ðyi ; EÞ
pffiffiffiffiffiffi0ffi
X i;
mn
¼
Dn;n0 pzffiffiffi
h 2
n0 ;
Z
1 p
dEz pffiffiffiffiffi nB ð
h! ÞGn0 ðyi ; Ez ; E þ h ! Þ
Ez
p
þ nB ð
h! Þþ1 Gn0 ðyi ; Ez ; E
h! Þ
(219)
(222)
(223)
where
Ai;i ¼ E h2 k2z
h2
2mnz mny y2
En ðyi Þ n ðyi ; kz ; EÞ
(224)
2
Ai1;i ¼
h
:
2mnz y2
(225)
The lead self-energies are
1
D2 kT
Deln;n0 ¼ ;0 þ b;b0 A 2
2
v
#
"
D2g h
D2f h
1
i;
þð1b;b0 Þ
Dn;n0 ¼ ;0 þ
b;b0
:
2!g
2
!f (220)
(221)
Here represents the phonon modes; nB represents the
Bose factor for phonons in equilibrium; and ! , !g and
!f represent phonon frequency due to intraband, g-type,
and f -type scattering processes due to mode ,
respectively. DA , Dg , and Df are the deformation
potential for acoustic intraband, g-type, and f -type
processes. is the mass density, k is the Boltzmann
constant, T is the temperature, and v is the velocity of
sound. The values for the above quantities are taken
from [13]. b and b0 are indexes representing valleys. The
following scattering processes are included: acoustic
phonon scattering in the elastic approximation and g-type
intervalley scattering with phonon energies of 12, 19, and
62 meV. We also remark that the imaginary part of the
electron–phonon self-energy, which is responsible for
scattering-induced broadening of energy levels, and
1540
2
h2
n;C ðy1 ; kz ; EÞ ¼
gs ðkz ; EÞ
2mnz y2
2
h2
n;C ðyN ; kz ; EÞ ¼
gd ðkz ; EÞ
2mnz y2
in
n;C ðy1 ; kz ; EÞ ¼ 2Im n;C ðy1 ; kz ; EÞ fs ðEÞ
(226)
(227)
¼ s fs ðEÞ
¼ 2Im n;C ðyN ; kz ; EÞ fd ðEÞ
(228)
in
n;C ðyN ; kz ; EÞ
¼ d fd ðEÞ
¼ 2Im n;C ðy1 ; kz ; EÞ ½1 fs ðEÞ
(229)
out
n;C ðy1 ; kz ; EÞ
out
n;C ðyN ; kz ; EÞ
¼ s ½1 fs ðEÞ
(230)
¼ 2Im n;C ðyN ; kz ; EÞ ½1 fd ðEÞ
¼ d ½1 fd ðEÞ
(231)
where y1 and yN are the leftmost (source-end) and
rightmost (drain-end) grid points, respectively; gs ðkz ; EÞ
and gd ðkz ; EÞ are the surface Green’s functions of the
source and drain leads, respectively; and fs and fd are the
Fermi functions in the source and drain leads, respectively.
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
The electron and current densities per energy given by
(128) and (149) can be simplified to
nn ðyi ; kz ; EÞ ¼ Gnn ðyi ; yi ; kz ; EÞ
ieX h2 n
Jn ðyi ; kz ; EÞ ¼
G ðyi ; yiþ1 ; kz ; EÞ
h n 2mny y2 n
Gnn ðyiþ1 ; yi ; kz ; EÞ :
(232)
(233)
The total electron and current densities at grid point yi
are given by
pffiffiffiffiffiffi0ffi
mn
pzffiffiffi
nðyi Þ ¼ 4
h 2
n Z
Z
dE
1
dEz pffiffiffiffiffi nn ðyi ; Ez ; EÞ
2
Ez
pffiffiffiffiffiffi0ffi
n
X m
pzffiffiffi
Jðyi Þ ¼ 4
h
2
n
Z
Z
dE
1
dEz pffiffiffiffiffi Jn ðyi ; Ez ; EÞ
2
Ez
X
(234)
(235)
where the prefactor of four in the above two equations
accounts for twofold spin and valley degeneracies. The
nonequilibrium electron and current densities are calculated in both the channel and extension regions using the
algorithm for Gn presented in Appendix II.
The Green’s function and Poisson’s equations are
solved self-consistently in the following way. The electrostatic potential is calculated by solving a 2-D Poisson
equation using fixed boundary at the gate leads and
floating boundary ðdVðyÞ=dyÞ ¼ 0 at the source and drain
leads [19]. The applied drain bias corresponds to a
difference in the Fermi levels used in the source and
drain regions. This potential is used in the mode space
calculation to determine the 1-D charge density nðyÞ.
Finally, the three-dimensional charge density is determined from
nðxi ; yi ; kz ; EÞ ¼ nn ðyi ; kz ; EÞjn ðxi ; yi Þj2 :
B. Results
Using the equations presented in the previous section,
we show results illustrating the role of scattering along the
channel length of a nanotransistor. First, we show that in
devices where the scattering length is comparable to the
channel length, the nanotransistor drive current is affected
by scattering at all points in the channel. Secondly, we
show that when hot electrons enter the drain extension
region of a nanotransistor, the drain extension region
cannot be modeled as a series resistance. Instead, the drain
extension should be included as part of the nonequilibrium
simulation region.
The device considered has a channel length of 25 nm,
body thickness of 1.5 nm, oxide thickness of 1.5 nm,
doping of 1 1020 cm3 in the source and drain
extension regions, and an intrinsic channel. Scattering
is included only from the source end of the channel
(5 nm) to the right boundary of scattering YRScatt by
setting the deformation potential in (220) and (221) to
zero to the right of YRScatt . The scattering lengths are
decreased by a factor of by modifying the deformation potential in (220)
pffiffiffiffi and (221) by an overall
multiplicative factor of .
Fig. 20 plots the drain current as a function of
ðYRScatt Þ. When the scattering length due to electron–
phonon interaction is 11 nm, the drive current degradation
is 30% due to scattering in the right half of the channel.
But when the scattering length is smaller (2.2 nm), the
drive current degradation is 15%. Thus the scattering in
the right half of the channel is important.
To illustrate the drain current degradation due to
scattering, we plot the current density Jðy; EÞ as a
(236)
The validity of the mode space approach has been tested
by [27]. It can be shown that the mode space approach is
valid when the wave function n ðx; yÞ at various y crosssections in (212) satisfies ðdn ðx; yÞ=dyÞ 0. That is, the
shape of the wave function at each cross-section should
not change significantly along the transport direction.
This implies that intersubband scattering due to changes
in potential profile are absent. This approximation seems
to be valid for channel thickness of less than 5 nm in
silicon [27].
Fig. 20. Drain current versus YRScatt for two different scattering
lengths. The channel length is 25 nm. (From [24].)
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Anantram et al.: Modeling of Nanoscale Devices
the reflection, and thereby the source injection barrier,
leads to a decrease in drain current [14].
To gain further insight into the role of scattering, we now
plot Jðy; EÞ for a scattering length of 2.2 nm [Fig. 21(b)],
which is five times smaller than the case of 11 nm [Fig. 21(a)].
As the channel length (25 nm) is much larger than 2.2 nm,
multiple scattering events now lead to an energy distribution
of current (in the right half of the channel) that is peaked
well below the source injection barrier.
The first moment of energy with respect to the current
distribution
function, which is defined as the ratio of
R
EJðy; EÞdE to total current, is also shown in Fig. 21 by the
dotted line with circles. This quantity is the mean energy at
which current flows. When the scattering length is much
smaller than the channel length, Rthe carriers relax
classically such that the first moment EJðy; EÞdE closely
tracks the potential profile, as seen in Fig. 21(b).
To further demonstrate the use of NEGF simulations,
we study the role of scattering by assuming that the
extension regions can be modeled as a classical series
resistance. Within the classical series resistance picture,
the current with scattering ðIscatt
Þ can be related to the
D
current without scattering ðInoscatt
Þ by [26]
D
ðVD Þ Inoscatt
ðVD VD Þ
Iscatt
D
D
(237)
where we have assumed that the source extension region
and device do not experience scattering. The potential drop
in the drain region within the classical series resistance
picture is VD ¼ Iscatt
ðVD ÞRD . In Fig. 22, the values of the
D
Fig. 21. Solid lines represent Jðy; EÞ for y equal to 17.5, 12.5, 7.5,
2.5, 2.5, 7.5, 12.5, and 17.5 nm, respectively, when scattering is
included every where in the channel. The dashed lines are the first
resonant level ðE1 Þ along the channel. The dotted lines represent the
first moment of energy with respect to the current distribution
R
function EJðy; EÞdE. (a) and (b) correspond to Lscatt ¼ 11 and 2.2 nm,
respectively. (From [24].)
function of energy at different positions y along the
channel (Fig. 21). Jðy; EÞ shows the energetic redistribution of carriers along the channel. When the scattering
length is 11 nm, which is comparable to the channel
length, Jðy; EÞ in the right-half of the channel is peaked in
energy above the source injection barrier, as shown in
Fig. 21(a). Scattering causes reflection of these energetic
electrons toward the source. These reflected electrons
lead to an increase in the channel electron density
(classical MOSFET electrostatics). As the charge in the
channel should be approximately Cox ðVG VS Þ, the
source injection barrier floats to a higher potential energy
to compensate for the reflected electrons. The increase in
1542
Fig. 22. The Id ðVd Þ plot in the ballistic limit is shown. Marked on this
plot are the drive current at Vd ¼ 0.6 in the ballistic limit, using the
series resistance picture and with scattering included using the NEGF
calculations. It should be noted that all three currents marked by the
arrows are calculated at Vd ¼ 0.6 V. (From [24].)
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
drain current versus drain voltage in the ballistic limit are
plotted. The currents in the ballistic limit, using the
classical series resistance picture and with the NEGF
method including scattering, are all marked by arrows.
Note that these three values of currents are all calculated at
Vd ¼ 0:6 V. It is seen from this plot that the current with
the NEGF method with scattering is significantly lower
than that obtained via the series resistance picture. Clearly,
the series resistance picture underestimates the detrimental nature of scattering in the drain end. The physics of the
large reduction in drain current was discussed in the
context of Fig. 21: When scattering in the channel does not
effectively thermalize carriers, the current distribution is
peaked at energies above the source injection barrier, upon
carriers exiting the channel. Scattering in the drain
extension region then causes reflection of electrons toward
the source-end. As a result, the source injection increases
so as to keep the electron density in the channel
approximately at Cox ðVG VS Þ. The drain current decreases dramatically as a result of the increase in source
injection barrier height.
IX. DISCUSSION AND SUMMARY
Our objectives in this paper have been to:
i) review the underlying assumptions of the traditional, semiclassical treatment of carrier transport
in semiconductor devices;
ii) describe how the semiclassical approach can be
applied to ballistic transport;
iii) discuss the Landauer–Buttiker approach to quantum transport in the phase coherent limit;
iv) introduce important elements of the non equilibrium Green’s function approach using Schrodinger’s
equation as a starting point;
v) demonstrate the application of the NEGF method
to the MOSFET in the ballistic limit and with
electron–phonon scattering.
It is appropriate to make a few comments about the
computational burden of the various transport models. One
reason that device engineers continue to use drift-diffusion
simulations rather than the more rigorous Monte Carlo
simulations is the enormous difference in computational
burden. For semiclassical transport, the fundamental
quantity is the carrier distribution function f ð~
r; ~
k; tÞ. To
find f ð~
r; ~
k Þ, we must solve the BTE, which is a sixdimensional equation. The difficulty of solving this sixdimensional equation is one reason that engineers continue
to rely on simplified models. For quantum transport, we can
take the Green’s function Gn ð~
r;~
r 0 ; EÞ as the fundamental
quantity. The Green’s function is a correlation function that
describes the phase relationship between the wavefunction
at~
r and~
r 0 for an electron injected at energy E. The quantum
transport problem is seven-dimensional, which makes it
much harder than the semiclassical problem. We can think
of ~
r and ~
r 0 as analogous to ~
r and ~
k in the semiclassical
approach, but there is no E in the semiclassical approach.
The reason is that, for a bulk semiconductor or in a device in
which the potential changes slowly, there is a relation
between E and ~
k, as determined by the semiconductor
bandstructure Eð~
k Þ. When the potential varies rapidly,
however, there is no Eð~
k Þ, and energy becomes a separate
dimension. Analysis of electronic devices by quantum
simulation is, however, becoming practical because device
dimensions are shrinking, which reduces the size of the
problem. Quantum simulations are also essential to
accurately model devices whose dimensions are comparable
to the phase-breaking length, and rely on tunneling and
wave interference for operation. The resonant tunneling
diode is the most successful example in this category.
We finally remark that the quantum mechanical
modeling outlined in this paper reduces to semiclassical
modeling when the device dimensions are much longer than
the phase-breaking scattering length. The transition from
the quantum to semiclassical regime was theoretically
addressed by [10] and [15], which derived the Boltzmann
transport equation starting from the nonequilibrium Green’s
function method. Similarly, the transition from the Wigner
function to drift-diffusion equations has been established in
[2]. The transition from quantum to classical transport in the
context of our discussion occurs when the self-energy
Phonon in (105), which represents electron–phonon
interaction, is nonzero. Electron–phonon interaction causes
reflection and breaks the phase-coherent evolution of
electron waves incident from the contacts. Apart from
this, electron–phonon interaction causes energy dissipation
in the device. These features result in a transition from
quantum mechanical to classical behavior. h
APPENDIX I
DYSON’S EQUATION FOR LAYERED
STRUCTURES
Partition the device layers into two regions Z and Z0 as
shown in Fig. 23. Dyson’s equation is a very useful
method that relates the Green’s function of the full
system Z þ Z0 in terms of the subsystems Z, Z0 and the
coupling between Z and Z0 . We will see below that, from a
computational point of view, Dyson’s equation provides
us with a systematic framework to calculate the diagonal
blocks of G and Gn without full inversion of the A matrix.
The reader should note that Dyson’s equation has a
significantly more general validity than implied in our
application here [16].
A. Dyson’s Equation for G
The Green’s function equation over the device
layers [(122)]
AG ¼ I
(A1)
GA ¼ I
(A2)
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Anantram et al.: Modeling of Nanoscale Devices
Fig. 23. Scheme of device for application of Dyson’s equation by splitting the device in two parts. (Adapted from [23].)
The Hermitian conjugate Green’s function ðGy Þ is by
definition related to G by
can be written as
AZ;Z
AZ0 ;Z
AZ;Z0
AZ0 ;Z0
GZ;Z0
GZ0 ;Z0
GZ;Z
GZ0 ;Z
¼
O
:
I
I
O
(A3)
Gy ¼ Gy0 þ Gy0 U y Gy
y0
(A9)
y y y0
¼G þ G U G :
(A10)
The solution of (A1) is
0
0
G ¼ G þ G UG
0
(A4)
0
¼ G þ GUG
(A5)
where
G¼
0
G ¼
U¼
GZ;Z
GZ;Z0
GZ0 ;Z
GZ0 ;Z0
G0Z;Z
O
O
G0Z0 ;Z0
Equation (A4) is Dyson’s equation for the Green’s
function.
B. Dyson’s Equation for Gn
The electron correlation function equation over the
device layers [(131)]
AGn ¼ in Gy
!
¼
O
AZ;Z0
AZ0 ;Z
O
A1
Z;Z
O
O
A1
Z0 ;Z0
:
!
can be written as
(A6)
It is verified by direct substitution of solutions (A5) and
(A4) to (A1) and (A2), respectively, and then, using the
above defintions
ðA þ UÞG0 ¼ I
AZ;Z
AZ;Z0
AZ0 ;Z
AZ0 ;Z0
¼
GnZ;Z
GnZ;Z0
GnZ0 ;Z
GnZ0 ;Z0
in
Z;Z
in
Z;Z0
in
Z0 ;Z
in
Z0 ;Z0
!
!
GyZ;Z
GyZ;Z0
GyZ0 ;Z
GyZ0 ;Z0
G ðA þ UÞ ¼ I:
: (A12)
The solution of (A11) is
Gn ¼ G0 UGn þ G0 in Gy
0
!
(A7)
or
1544
(A11)
(A8)
(A13)
where G0 and U have been defined in (A6). Functions Gn
and Gy are readily defined by (A11) and (A10), respectively.
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
Using Gy ¼ Gy0 þ Gy0 U y Gy , (A13) can be written as
Gn ¼ Gn0 þ Gn0 U y Gy þ G0 UGn
(A14)
n y y0
¼ G þ GUG þ G U G
(A15)
Gn0 ¼ G0 in Gy0 :
(A16)
n0
n0
where
A PP E NDI X I I
ALGORITHM TO CALCULATE G AND Gn
Why algorithm: A typical simulation of a nanoelectronic device consists of solving Poisson’s equation selfconsistently with the Green’s function equations. The
input to Poisson’s equation is the charge density, which is
obtained by integrating Gnq;q ðEÞ over energy. The index q
here runs over the layers of the device. In order to
calculate the current density, one requires the elements
Gnq;qþ1 ðEÞ from the diagonals adjacent to the main
diagonal. That is, we do not require the entire Gn matrix
in most situations. Instead, just the three diagonals of
elements or of blocks of elements need to be calculated.
This also applies for the hole correlation function Gp .
Provided that Nx is the dimension of the Hamiltonian
of each layer and N is the total number of layers, the size of
the matrix A equals Nx N. The operation count for the full
matrix inversion G ¼ A1 is proportional to ðNx NÞ3 . The
computational cost of obtaining the diagonal elements of
the Gn matrix at each energy is approximately Nx3 N 3
operations if Gn ¼ Gin Gy is used. Therefore, it is highly
desirable to find less expensive algorithms that avoid full
inversion of matrix A and specifically take advantage of the
fact that only the diagonal elements of Green’s functions
are required. Another reason to prefer such algorithms is
the memory storage. If one had had to retain the whole
matrix G in the memory, it might had not fit into the onchip cache and had required using slower access memory
(RAM or a swap file). That would have significantly slowed
down the calculations.
One such algorithm to calculate Gn and Gp that is valid
for the block tridiagonal form of matrix A developed in [23]
is presented in this section. The algorithm to calculate the
diagonal blocks of G was developed in [12]. The operation
count of this algorithm scales approximately as Nx3 N. The
dependence on Nx3 arises because matrices of dimension of
the sub-Hamiltonian of layers should be inverted, and the
dependence on N corresponds to one such inversion for
each of the N layers.
The algorithm consists of two steps. In the first step,
the diagonal blocks of the left connected and full Green’s
function are evaluated. In the second step, these results are
used to evaluate the diagonal blocks of the Gn Green’s
function.
A. Recursive Algorithm for G
i) Left-connected Green’s function (Fig. 24): The
left-connected (superscript L) Green’s function g Lq
is defined by the first q blocks of (122) by [12]
A1:q;1:q g Lq ¼ Iq;q
(B1)
Fig. 24. Illustration for the relation between the left-connected Green’s functions for adjacent layers. (Adapted from [23].)
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Anantram et al.: Modeling of Nanoscale Devices
g nLqþ1 is defined in a manner identical to g nLq
except that the left-connected system is composed
of the first q þ 1 blocks of (131).
The equation governing g nLqþ1 follows from (A12)
by setting Z ¼ 1 : q and Z0 ¼ q þ 1. Using the
Dyson’s equations for G [(A4)] and Gn [(A14)],
nLqþ1
gqþ1;qþ1 is recursively obtained as
where we introduce a shorthand Iq;q ¼ I1:q;1:q . The
matrix g Lqþ1 is defined in a manner identical to g Lq
except that the left-connected system is composed
of the first q þ 1 blocks of (122). In terms of (A3),
the equation governing g Lqþ1 can be expressed via
the solution g Lq by setting Z ¼ 1 : q and Z0 ¼ q þ 1.
Using Dyson’s equation [(A4)], we obtain
1
Lqþ1
Lq
Aq;qþ1
: (B2)
gqþ1;qþ1 ¼ Aqþ1;qþ1 Aqþ1;q gq;q
ii)
Note that the last element of this progression g LN is
equal to the fully connected Green’s function G,
which is the solution to (122).
Full Green’s function in terms of the left-connected
Green’s function: Consider the special case of (A3)
in which AZ;Z ¼ A1:q;1:q , AZ0 ;Z0 ¼ Aqþ1:N;qþ1:N , and
AZ;Z0 ¼ A1:q;qþ1:N . Noting that the only nonzero
element of A1:q;qþ1:N is Aq;qþ1 and using (A4), we
obtain
(B4)
The equations for the adjacent diagonals are
obtained similarly as
nLq
nLq y
Lq
gq;q
Aq;qþ1 Gyqþ1;q gq;q
Aq;qþ1 Gnqþ1;q : (B9)
Gnq;q ¼ gq;q
Lq
Lq
gq;q
Aq;qþ1 Gqþ1;q :
¼ gq;q
Lq
Gqþ1;q ¼ Gqþ1;qþ1 Aqþ1;q gq;q
Gq;qþ1 ¼
Lq
gq;q
Aq;qþ1 Gqþ1;qþ1 :
(B3)
(B5)
(B6)
Both Gq;q and Gqþ1;q are used in the algorithm for electron
density, and so storing both sets of matrices is necessary.
Making use of the above equations, the algorithm to
obtain the three diagonals of G is as follows.
L1
1) g11
¼ A1
11 .
2) For q ¼ 1; 2; . . . ; N 1, compute (B2).
Lq y
3) For q ¼ 1; 2; . . . ; N, compute ðgqq
Þ.
Lq
4) GN;N ¼ gq;q
.
5) For q ¼ N 1; N 2; . . . ; 1, compute (B5), (B6),
and (B4) (in this order).
6) For q ¼ 1; 2; . . . ; N, compute ðGq;qþ1 Þy and
ðGqþ1;q Þy .
B. Recursive Algorithm for Gn
Left-connected Gn (Fig. 24): The function g nLq is
the counterpart of g Lq and is defined by the first q
blocks of (131) [23]
yLq
A1:q;1:q g nLq ¼ in
1:q;1:q g1:q;1:q :
1546
(B8)
nLq y
where in
qþ1 ¼ Aqþ1;q gq;q Aq;qþ1 . Equation (B8)
nLqþ1
has the physical meaning that gqþ1;qþ1 has
contributions due to an effective self-energy
due to the left-connected structure that ends at
q, which is represented by in
qþ1 and the diagonal
self-energy component at grid point q þ 1 (in
DD
of (131)).
Full electron correlation function in terms of
left-connected Green’s function: Consider (A12)
such that AZ ¼ A1:q;1:q , A0Z ¼ Aqþ1:N;qþ1:N and
AZ;Z0 ¼ A1:q;qþ1:N . Noting that the only non zero
element of A1:q;qþ1:N is Aq;qþ1 and using (A14), we
obtain
Lq
Lq
Lq
þ gq;q
Aq;qþ1 Gqþ1;qþ1 Aqþ1;q gq;q
Gq;q ¼ gq;q
i)
h
i
nLqþ1
Lqþ1
Lqþ1y
in
gqþ1;qþ1 ¼ gqþ1;qþ1 in
þ
qþ1;qþ1
qþ1 gqþ1;qþ1
(B7)
Proceedings of the IEEE | Vol. 96, No. 9, September 2008
ii)
Using (A15), Gnqþ1;q can be written in terms of
Gnqþ1;qþ1 and other known Green’s functions as
nLq
yLq
Gnqþ1;qþ1 Ayqþ1;q gq;q
:
Gnqþ1;q ¼ Gqþ1;qþ1 Aqþ1;q gq;q
(B10)
Substituting (B10) in (B9) and using (A4) and
(A5), we obtain
nLq
Lq
yLq
Gnq;q ¼ gq;q
þ gq;q
Aq;qþ1 Gnqþ1;qþ1 Ayqþ1;q gq;q
h
i
nLq y
nLq
gq;q
Aq;qþ1 Gyqþ1;q þ Gq;qþ1 Aqþ1;q gq;q
: (B11)
The terms inside the square brackets of (B11) are
Hermitian conjugates of each other. In view of the
above equations, the algorithm to compute the
diagonal blocks of Gn and Gp is given by the following steps.
nL1
L1 in L1y
1) g11
¼ g11
11 g11 .
2) For q ¼ 1; 2; . . . ; N 1, compute (B8).
nLN
3) GnN;N ¼ gNN
.
Anantram et al.: Modeling of Nanoscale Devices
For q ¼ N 1; N 2; . . . ; 1, compute (B11)
and (B10).
5) Use Gnq;qþ1 ¼ ðGnqþ1;q Þy .
6) Use Gp ¼ iðG Gy Þ Gn .
The above algorithm is illustrated by a Matlab code in
Appendix III.
Challenging problem: The algorithm presented here
solves for the three block diagonals of G, Gn , and Gp .
Each of n blocks on the main diagonal corresponds to a
layer of the device. All blocks in the three diagonals are
treated as a full matrix. It is highly desirable to find a
more efficient algorithm that finds only the diagonal
4)
elements within each block rather than complete
blocks.
APPENDIX III
CODE OF THE RECURSIVE ALGORITHM
The listing of Matlab code recursealg3d.m is provided
here for illustration of the algorithm described in
Appendix II. Open-source simulation tools written in
Matlab and using this algorithm are nanoMOS [19], [20],
for a 2-D semiconductor transistor, and MOSCNT [11], for
a carbon nanotube transistor.
function [Grl,Grd,Gru,Gnl,Gnd,Gnu,Gpl,Gpd,Gpu,grL,ginL] . . .
= recursealg3d(Np,Al,Ad,Au,Sigin,Sigout)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function [Grl,Grd,Gru,Gnl,Gnd,Gnu,Gpl,Gpd,Gpu,grL,ginL] . . .
% = recursealg3d(Np,Al,Ad,Au,Sigin,Sigout)
% recursive algorithm to solve for the diagonal elements of
% the nonequilibrium Green’s function
% ‘‘l’’ ¼ lower diagonal, size (1, Np-1)=[G(2,1) . . . G(end,end-1)]
% ‘‘d’’ ¼ main diagonal, size ð1; NpÞ ¼ ½Gð1; 1Þ . . . Gðend; endÞ
% ‘‘u’’ ¼ upper diagonal, size ð1; Np 1Þ ¼ ½Gð1; 2Þ . . . : Gðend 1; endÞ
% Grl, Grd, Gru=retarded Green’s function
% Gnl, Gnd, Gnu=electron Green’s function
% Gpl, Gpd, Gpu=hole Green’s function
% grL=left-connected Green’s function
% ginL=left-connected in-scattering function
% Np= size of the matrices
% Al, Ad, Au= matrix of coefficients
% Sigin= matrix of in-scattering self-energies (diagonal)
% Sigout= matrix of out-scattering self-energies (diagonal)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dmitri Nikonov, Intel Corp. and Siyu Koswatta, Purdue University, 2004
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
flag Gp ¼ ‘no’;
Al cr ¼ conjðAuÞ;
Ad cr ¼ conjðAdÞ;
% Hermitian conjugate of the coefficient matrix
Au cr ¼ conjðAlÞ;
grL ¼ zerosð1; NpÞ;
% initialize left-connected function
ginL ¼ zerosð1; NpÞ;
% initialize left-connected in-scattering function
gipL ¼ zerosð1; NpÞ;
% initialize left-connected out-scattering function
Grl ¼ zerosð1; Np 1Þ;
Grd ¼ zerosð1; NpÞ;
% initialize the Green’s function
Gru ¼ zerosð1; Np 1Þ;
Gnl ¼ zerosð1; Np 1Þ;
Gnd ¼ zerosð1; NpÞ;
% initialize the electron coherence function
Gnu ¼ zerosð1; Np 1Þ;
Gpl ¼ zerosð1; Np 1Þ;
Gpd ¼ zerosð1; NpÞ;
% initialize the hole coherence function
Gpu ¼ zerosð1; Np 1Þ;
grLð1Þ ¼ 1=Adð1Þ;
% step 1
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
1547
Anantram et al.: Modeling of Nanoscale Devices
for q ¼ 2 : Np
% obtain the left-connected function
grLðqÞ ¼ 1=ðAd ðqÞ Alðq 1Þ grLðq 1Þ Auðq 1ÞÞ;
end
gaL ¼ conjðgrLÞ;
% advanced left-connected function
GrdðNpÞ ¼ grLðNpÞ;
% step 2
for q ¼ ðNp 1Þ : 1 : 1
GrlðqÞ ¼ Grdðq þ 1Þ AlðqÞ grLðqÞ;
% obtain the subdiagonal of the Green’s function
GruðqÞ ¼ grLðqÞ AuðqÞ Grdðq þ 1Þ;
% obtain the super-diagonal of the Green’s function
GrdðqÞ ¼ grLðqÞ grLðqÞ AuðqÞ GrlðqÞ;
% obtain the diagonal of the Green’s function
end
Gal ¼ conjðGruÞ;
Gad ¼ conjðGrdÞ;
% advanced Green’s function
Gau ¼ conjðGalÞ;
ginLð1Þ ¼ grLð1Þ Siginð1Þ gaLð1Þ;
% step 3
for q ¼ 2 : Np
sla2 ¼ Alðq 1Þ ginLðq 1Þ Au crðq 1Þ;
prom ¼ SiginðqÞ þ sla2;
ginLðqÞ ¼ realðgrLðqÞ prom gaLðqÞÞ;
% left-connected in-scattering function
end
GndðNpÞ ¼ ginLðNpÞ;
% step 4
for q ¼ ðNp 1Þ : 1 : 1
GnlðqÞ ¼ Grdðq þ 1Þ AlðqÞ ginLðqÞ Gndðq þ 1Þ Al crðqÞ gaLðqÞ;
% obtain the lower diagonal of the electron Green’s function
GndðqÞ ¼ realðginLðqÞ þ grLðqÞ AuðqÞ Gndðq þ 1Þ Al crðqÞ gaLðqÞ . . .
ðginLðqÞ Au crðqÞ GalðqÞþ GruðqÞ AlðqÞ ginLðqÞ ÞÞ;
end
Gnu ¼ conjðGnlÞ;
% upper diagonal of the electron function
switch flag_Gp
case Fyes_
gipLð1Þ ¼ grLð1Þ Sigoutð1Þ gaLð1Þ;
% step 5
for q ¼ 2 : Np
sla2 ¼ Alðq 1Þ gipLðq 1Þ Au crðq 1Þ;
prom ¼ SigoutðqÞ þ sla2;
gipLðqÞ ¼ realðgrLðqÞ prom gaLðqÞÞ;
% left-connected out-scattering function
end
GpdðNpÞ ¼ gipLðNpÞ;
% step 6
for q ¼ ðNp 1Þ : 1 : 1
GplðqÞ ¼ Grdðq þ 1Þ AlðqÞ gipLðqÞ Gndðq þ 1Þ Al crðqÞ gaLðqÞ;
% obtain the lower diagonal of the hole Green’s function
GpdðqÞ ¼ realðgipLðqÞ þ grLðqÞ AuðqÞ Gpdðq þ 1Þ Al crðqÞ gaLðqÞ . . .
ðgipLðqÞ Au crðqÞ GalðqÞ þ GruðqÞ AlðqÞ gipLðqÞ ÞÞ;
end
Gpu ¼ conjðGplÞ;
% upper diagonal of the hole function
case Fno_
Gpl ¼ i ðGrl GalÞ Gnl;
Gpd ¼ i ðGrd GadÞ Gnd;
% hole Green’s function
Gpu ¼ i ðGru GauÞ Gnu;
end
Gnd ¼ realðGndÞ;
Gpd ¼ realðGpdÞ;
jnzer ¼ findðGndG0Þ;
GndðjnzerÞ ¼ 0;
jpzer ¼ findðGpdG0Þ;
GpdðjpzerÞ ¼ 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Proceedings of the IEEE | Vol. 96, No. 9, September 2008
Anantram et al.: Modeling of Nanoscale Devices
Acknowledgment
The authors would like to thank S. Datta for many
useful and inspirational discussions on the NEGF method.
M. S. Lundstrom is indebted to his students who have, over
the past several years, applied the NEGF methods to
numerous problems in nanotransistors. Anantram is
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ABOUT THE AUTHORS
M. P. Anantram received the B.Sc. degree in
applied sciences from the PSG College of Technology, Coimbatore, India, in 1986, the M.Sc.
degree in physics from the University of Poona,
Pune, India, in 1989, and the Ph.D. degree in
electrical engineering from Purdue University,
West Lafayette, IN, in 1995.
He was the Group Lead for Computational
Nanoelectronics and, since 2004, has been a
Fellow with the University Affiliated Research
Center, National Aeronautics and Space Administration, Ames Research
Center, where his group developed the first 2-D quantum simulator for
nanotransistors and an algorithm to compute the charge and current
density in nanodevices. He has also worked extensively on the modeling
and prediction of electrical and electromechanical properties of nanostructures. Since 2006, he has been a Professor with the Nanotechnology
Engineering Group, Department of Electrical and Computer Engineering,
University of Waterloo, Waterloo, ON, Canada. His research encompasses
theory and computational modeling of semiconductor and molecular
nanodevices. He serves on the program and organizing committees of
nanotechnology conferences.
Dr. Anantram is an Associate Editor of the IEEE TRANSACTIONS ON
NANOTECHNOLOGY and the Education Chair of the IEEE Nanotechnology
Council. He has received best paper awards for his work on nanotransistors and electromechanical properties of nanotubes. He was a corecipient of the Highest Technical Achievement in Applied Sciences from
the Computer Sciences Corporation in 2003 for his work on theory and
modeling of nanodevices.
Vol. 96, No. 9, September 2008 | Proceedings of the IEEE
1549
Anantram et al.: Modeling of Nanoscale Devices
Mark S. Lundstrom (Fellow, IEEE) received the
B.E.E. and M.S.E.E. degrees from the University of
Minnesota, Minneapolis, in 1973 and 1974, respectively, and the Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN,
in 1980.
From 1974 to 1977, he was with the HewlettPackard Corporation, Loveland, CO, working on
integrated circuit process development and manufacturing support. In 1980, he joined the School
of Electrical and Computer Engineering, Purdue University, where he is
currently the Don and Carol Scifres Distinguished Professor of Electrical
and Computer Engineering and the founding Director of the Network for
Computational Nanotechnology. From 1989 to 1993, he was a Director of
the Optoelectronics Research Center, Purdue University, and, from 1991
to 1994, the Assistant Dean of Engineering. His current research interests
include the physics of small electronic devices, especially nanoscale
transistors, and carrier transport in semiconductor devices.
Prof. Lundstrom is a Fellow of the American Physical Society and the
American Association for the Advancement of Science. He is a Distinguished Lecturer of the IEEE Electron Device Society. He received the
Frederick Emmons Terman Award from the American Society for
Engineering Education in 1992. He (with S. Datta) received the IEEE
Cledo Brunetti Award for their work on nanoscale electronic devices and
the Semiconductor Research Corporation’s Technical Excellence Award
in 2002. He received the Semiconductor Industry Association’s University
Researcher Award for his career contributions to the physics and
simulation of semiconductor devices in 2005. Most recently, in 2006,
he was the inaugural recipient of the IEEE Electron Device Society’s
Education Award.
1550
Dmitri E. Nikonov (Senior Member, IEEE) received the M.S. degree in aeromechanical engineering from the Moscow Institute of Physics and
Technology, Moscow, Russia, in 1992 and the Ph.D.
degree in physics from Texas A&M University,
College Station, in 1996.
While at Texas A&M he participated in the
demonstration of the world’s first laser without
population inversion. He joined Intel Corporation
in 1998, where he is presently a Project Manager
with the Technology Strategy Group, Santa Clara, CA. He is responsible
for managing joint research programs with universities on nanotechnology, optoelectronics, and advanced devices. From 1997 to 1998, he was a
Research Engineer and Lecturer with the Department of Electrical and
Computer Engineering, University of California, Santa Barbara. In
2006, he became an Adjunct Associate Professor of electrical and
computer engineering at Purdue University, West Lafayette, IN. He has
38 publications in refereed journals in quantum optics, free-electron,
gas and semiconductor lasers, nanoelectronics, spintronics, and
quantum devices simulation. He has received 29 patents in optoelectronics, integrated optics, and spintronic devices.
Dr. Nikonov was a Finalist for the Best Doctoral Thesis award from the
American Physical Society in 1997.
Proceedings of the IEEE | Vol. 96, No. 9, September 2008